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<div class="section" id="general-unconstrained-minimization">
<span id="chapter-gradient-problem-solver"></span><h1>General Unconstrained Minimization<a class="headerlink" href="#general-unconstrained-minimization" title="Permalink to this headline">¶</a></h1>
<div class="section" id="modeling">
<h2>Modeling<a class="headerlink" href="#modeling" title="Permalink to this headline">¶</a></h2>
<div class="section" id="firstorderfunction">
<h3><a class="reference internal" href="#_CPPv218FirstOrderFunction" title="FirstOrderFunction"><code class="xref cpp cpp-class docutils literal"><span class="pre">FirstOrderFunction</span></code></a><a class="headerlink" href="#firstorderfunction" title="Permalink to this headline">¶</a></h3>
<dl class="class">
<dt id="_CPPv218FirstOrderFunction">
<span id="FirstOrderFunction"></span><em class="property">class </em><code class="descclassname"></code><code class="descname">FirstOrderFunction</code><a class="headerlink" href="#_CPPv218FirstOrderFunction" title="Permalink to this definition">¶</a></dt>
<dd><p>Instances of <a class="reference internal" href="#_CPPv218FirstOrderFunction" title="FirstOrderFunction"><code class="xref cpp cpp-class docutils literal"><span class="pre">FirstOrderFunction</span></code></a> implement the evaluation of
a function and its gradient.</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">FirstOrderFunction</span> <span class="p">{</span>
<span class="k">public</span><span class="o">:</span>
<span class="k">virtual</span> <span class="o">~</span><span class="n">FirstOrderFunction</span><span class="p">()</span> <span class="p">{}</span>
<span class="k">virtual</span> <span class="kt">bool</span> <span class="n">Evaluate</span><span class="p">(</span><span class="k">const</span> <span class="kt">double</span><span class="o">*</span> <span class="k">const</span> <span class="n">parameters</span><span class="p">,</span>
<span class="kt">double</span><span class="o">*</span> <span class="n">cost</span><span class="p">,</span>
<span class="kt">double</span><span class="o">*</span> <span class="n">gradient</span><span class="p">)</span> <span class="k">const</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="k">virtual</span> <span class="kt">int</span> <span class="nf">NumParameters</span><span class="p">()</span> <span class="k">const</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="p">};</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK18FirstOrderFunction8EvaluateEPCKdPdPd">
<span id="FirstOrderFunction::Evaluate__doubleCPC.doubleP.doublePC"></span>bool <code class="descclassname">FirstOrderFunction::</code><code class="descname">Evaluate</code><span class="sig-paren">(</span><em class="property">const</em> double *<em class="property">const</em> <em>parameters</em>, double *<em>cost</em>, double *<em>gradient</em><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK18FirstOrderFunction8EvaluateEPCKdPdPd" title="Permalink to this definition">¶</a></dt>
<dd><p>Evaluate the cost/value of the function. If <code class="docutils literal"><span class="pre">gradient</span></code> is not
<code class="docutils literal"><span class="pre">NULL</span></code> then evaluate the gradient too. If evaluation is
successful return, <code class="docutils literal"><span class="pre">true</span></code> else return <code class="docutils literal"><span class="pre">false</span></code>.</p>
<p><code class="docutils literal"><span class="pre">cost</span></code> guaranteed to be never <code class="docutils literal"><span class="pre">NULL</span></code>, <code class="docutils literal"><span class="pre">gradient</span></code> can be <code class="docutils literal"><span class="pre">NULL</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK18FirstOrderFunction13NumParametersEv">
<span id="FirstOrderFunction::NumParametersC"></span>int <code class="descclassname">FirstOrderFunction::</code><code class="descname">NumParameters</code><span class="sig-paren">(</span><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK18FirstOrderFunction13NumParametersEv" title="Permalink to this definition">¶</a></dt>
<dd><p>Number of parameters in the domain of the function.</p>
</dd></dl>
</div>
<div class="section" id="gradientproblem">
<h3><a class="reference internal" href="#_CPPv215GradientProblem" title="GradientProblem"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblem</span></code></a><a class="headerlink" href="#gradientproblem" title="Permalink to this headline">¶</a></h3>
<dl class="class">
<dt id="_CPPv215GradientProblem">
<span id="GradientProblem"></span><em class="property">class </em><code class="descclassname"></code><code class="descname">GradientProblem</code><a class="headerlink" href="#_CPPv215GradientProblem" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">GradientProblem</span> <span class="p">{</span>
<span class="k">public</span><span class="o">:</span>
<span class="k">explicit</span> <span class="n">GradientProblem</span><span class="p">(</span><span class="n">FirstOrderFunction</span><span class="o">*</span> <span class="n">function</span><span class="p">);</span>
<span class="n">GradientProblem</span><span class="p">(</span><span class="n">FirstOrderFunction</span><span class="o">*</span> <span class="n">function</span><span class="p">,</span>
<span class="n">LocalParameterization</span><span class="o">*</span> <span class="n">parameterization</span><span class="p">);</span>
<span class="kt">int</span> <span class="nf">NumParameters</span><span class="p">()</span> <span class="k">const</span><span class="p">;</span>
<span class="kt">int</span> <span class="nf">NumLocalParameters</span><span class="p">()</span> <span class="k">const</span><span class="p">;</span>
<span class="kt">bool</span> <span class="nf">Evaluate</span><span class="p">(</span><span class="k">const</span> <span class="kt">double</span><span class="o">*</span> <span class="n">parameters</span><span class="p">,</span> <span class="kt">double</span><span class="o">*</span> <span class="n">cost</span><span class="p">,</span> <span class="kt">double</span><span class="o">*</span> <span class="n">gradient</span><span class="p">)</span> <span class="k">const</span><span class="p">;</span>
<span class="kt">bool</span> <span class="nf">Plus</span><span class="p">(</span><span class="k">const</span> <span class="kt">double</span><span class="o">*</span> <span class="n">x</span><span class="p">,</span> <span class="k">const</span> <span class="kt">double</span><span class="o">*</span> <span class="n">delta</span><span class="p">,</span> <span class="kt">double</span><span class="o">*</span> <span class="n">x_plus_delta</span><span class="p">)</span> <span class="k">const</span><span class="p">;</span>
<span class="p">};</span>
</pre></div>
</div>
<p>Instances of <a class="reference internal" href="#_CPPv215GradientProblem" title="GradientProblem"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblem</span></code></a> represent general non-linear
optimization problems that must be solved using just the value of the
objective function and its gradient. Unlike the <code class="xref cpp cpp-class docutils literal"><span class="pre">Problem</span></code>
class, which can only be used to model non-linear least squares
problems, instances of <a class="reference internal" href="#_CPPv215GradientProblem" title="GradientProblem"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblem</span></code></a> not restricted in the
form of the objective function.</p>
<p>Structurally <a class="reference internal" href="#_CPPv215GradientProblem" title="GradientProblem"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblem</span></code></a> is a composition of a
<a class="reference internal" href="#_CPPv218FirstOrderFunction" title="FirstOrderFunction"><code class="xref cpp cpp-class docutils literal"><span class="pre">FirstOrderFunction</span></code></a> and optionally a
<code class="xref cpp cpp-class docutils literal"><span class="pre">LocalParameterization</span></code>.</p>
<p>The <a class="reference internal" href="#_CPPv218FirstOrderFunction" title="FirstOrderFunction"><code class="xref cpp cpp-class docutils literal"><span class="pre">FirstOrderFunction</span></code></a> is responsible for evaluating the cost
and gradient of the objective function.</p>
<p>The <code class="xref cpp cpp-class docutils literal"><span class="pre">LocalParameterization</span></code> is responsible for going back and
forth between the ambient space and the local tangent space. When a
<code class="xref cpp cpp-class docutils literal"><span class="pre">LocalParameterization</span></code> is not provided, then the tangent space
is assumed to coincide with the ambient Euclidean space that the
gradient vector lives in.</p>
<p>The constructor takes ownership of the <a class="reference internal" href="#_CPPv218FirstOrderFunction" title="FirstOrderFunction"><code class="xref cpp cpp-class docutils literal"><span class="pre">FirstOrderFunction</span></code></a> and
<code class="xref cpp cpp-class docutils literal"><span class="pre">LocalParamterization</span></code> objects passed to it.</p>
<dl class="function">
<dt id="_CPPv25SolveRKN21GradientProblemSolver7OptionsERK15GradientProblemPdPN21GradientProblemSolver7SummaryE">
<span id="Solve__GradientProblemSolver::OptionsCR.GradientProblemCR.doubleP.GradientProblemSolver::SummaryP"></span>void <code class="descclassname"></code><code class="descname">Solve</code><span class="sig-paren">(</span><em class="property">const</em> GradientProblemSolver::<a class="reference internal" href="#_CPPv2N21GradientProblemSolver7OptionsE" title="GradientProblemSolver::Options">Options</a> &<em>options</em>, <em class="property">const</em> <a class="reference internal" href="#_CPPv215GradientProblem" title="GradientProblem">GradientProblem</a> &<em>problem</em>, double *<em>parameters</em>, GradientProblemSolver::<a class="reference internal" href="#_CPPv2N21GradientProblemSolver7SummaryE" title="GradientProblemSolver::Summary">Summary</a> *<em>summary</em><span class="sig-paren">)</span><a class="headerlink" href="#_CPPv25SolveRKN21GradientProblemSolver7OptionsERK15GradientProblemPdPN21GradientProblemSolver7SummaryE" title="Permalink to this definition">¶</a></dt>
<dd><p>Solve the given <a class="reference internal" href="#_CPPv215GradientProblem" title="GradientProblem"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblem</span></code></a> using the values in
<code class="docutils literal"><span class="pre">parameters</span></code> as the initial guess of the solution.</p>
</dd></dl>
</div>
</div>
<div class="section" id="solving">
<h2>Solving<a class="headerlink" href="#solving" title="Permalink to this headline">¶</a></h2>
<div class="section" id="gradientproblemsolver-options">
<h3><a class="reference internal" href="#_CPPv2N21GradientProblemSolver7OptionsE" title="GradientProblemSolver::Options"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblemSolver::Options</span></code></a><a class="headerlink" href="#gradientproblemsolver-options" title="Permalink to this headline">¶</a></h3>
<dl class="class">
<dt id="_CPPv2N21GradientProblemSolver7OptionsE">
<span id="GradientProblemSolver::Options"></span><em class="property">class </em><code class="descclassname">GradientProblemSolver::</code><code class="descname">Options</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7OptionsE" title="Permalink to this definition">¶</a></dt>
<dd><p><a class="reference internal" href="#_CPPv2N21GradientProblemSolver7OptionsE" title="GradientProblemSolver::Options"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblemSolver::Options</span></code></a> controls the overall
behavior of the solver. We list the various settings and their
default values below.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK21GradientProblemSolver7Options7IsValidEP6string">
<span id="GradientProblemSolver::Options::IsValid__stringPC"></span>bool <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">IsValid</code><span class="sig-paren">(</span>string *<em>error</em><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK21GradientProblemSolver7Options7IsValidEP6string" title="Permalink to this definition">¶</a></dt>
<dd><p>Validate the values in the options struct and returns true on
success. If there is a problem, the method returns false with
<code class="docutils literal"><span class="pre">error</span></code> containing a textual description of the cause.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options26line_search_direction_typeE">
<span id="GradientProblemSolver::Options::line_search_direction_type__LineSearchDirectionType"></span>LineSearchDirectionType <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">line_search_direction_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options26line_search_direction_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">LBFGS</span></code></p>
<p>Choices are <code class="docutils literal"><span class="pre">STEEPEST_DESCENT</span></code>, <code class="docutils literal"><span class="pre">NONLINEAR_CONJUGATE_GRADIENT</span></code>,
<code class="docutils literal"><span class="pre">BFGS</span></code> and <code class="docutils literal"><span class="pre">LBFGS</span></code>.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options16line_search_typeE">
<span id="GradientProblemSolver::Options::line_search_type__LineSearchType"></span>LineSearchType <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">line_search_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options16line_search_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">WOLFE</span></code></p>
<p>Choices are <code class="docutils literal"><span class="pre">ARMIJO</span></code> and <code class="docutils literal"><span class="pre">WOLFE</span></code> (strong Wolfe conditions).
Note that in order for the assumptions underlying the <code class="docutils literal"><span class="pre">BFGS</span></code> and
<code class="docutils literal"><span class="pre">LBFGS</span></code> line search direction algorithms to be guaranteed to be
satisifed, the <code class="docutils literal"><span class="pre">WOLFE</span></code> line search should be used.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options33nonlinear_conjugate_gradient_typeE">
<span id="GradientProblemSolver::Options::nonlinear_conjugate_gradient_type__NonlinearConjugateGradientType"></span>NonlinearConjugateGradientType <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">nonlinear_conjugate_gradient_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options33nonlinear_conjugate_gradient_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">FLETCHER_REEVES</span></code></p>
<p>Choices are <code class="docutils literal"><span class="pre">FLETCHER_REEVES</span></code>, <code class="docutils literal"><span class="pre">POLAK_RIBIERE</span></code> and
<code class="docutils literal"><span class="pre">HESTENES_STIEFEL</span></code>.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options13max_lbfs_rankE">
<span id="GradientProblemSolver::Options::max_lbfs_rank__i"></span>int <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_lbfs_rank</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options13max_lbfs_rankE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: 20</p>
<p>The L-BFGS hessian approximation is a low rank approximation to the
inverse of the Hessian matrix. The rank of the approximation
determines (linearly) the space and time complexity of using the
approximation. Higher the rank, the better is the quality of the
approximation. The increase in quality is however is bounded for a
number of reasons.</p>
<blockquote>
<div><ol class="arabic simple">
<li>The method only uses secant information and not actual
derivatives.</li>
<li>The Hessian approximation is constrained to be positive
definite.</li>
</ol>
</div></blockquote>
<p>So increasing this rank to a large number will cost time and space
complexity without the corresponding increase in solution
quality. There are no hard and fast rules for choosing the maximum
rank. The best choice usually requires some problem specific
experimentation.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options39use_approximate_eigenvalue_bfgs_scalingE">
<span id="GradientProblemSolver::Options::use_approximate_eigenvalue_bfgs_scaling__b"></span>bool <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">use_approximate_eigenvalue_bfgs_scaling</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options39use_approximate_eigenvalue_bfgs_scalingE" title="Permalink to this definition">¶</a></dt>
<dd><blockquote>
<div><p>Default: <code class="docutils literal"><span class="pre">false</span></code></p>
<p>As part of the <code class="docutils literal"><span class="pre">BFGS</span></code> update step / <code class="docutils literal"><span class="pre">LBFGS</span></code> right-multiply
step, the initial inverse Hessian approximation is taken to be the
Identity. However, <a class="reference internal" href="bibliography.html#oren" id="id1">[Oren]</a> showed that using instead <span class="math">\(I *
\gamma\)</span>, where <span class="math">\(\gamma\)</span> is a scalar chosen to approximate an
eigenvalue of the true inverse Hessian can result in improved
convergence in a wide variety of cases. Setting
<code class="docutils literal"><span class="pre">use_approximate_eigenvalue_bfgs_scaling</span></code> to true enables this
scaling in <code class="docutils literal"><span class="pre">BFGS</span></code> (before first iteration) and <code class="docutils literal"><span class="pre">LBFGS</span></code> (at each
iteration).</p>
<p>Precisely, approximate eigenvalue scaling equates to</p>
<div class="math">
\[\gamma = \frac{y_k' s_k}{y_k' y_k}\]</div>
<p>With:</p>
</div></blockquote>
<div class="math">
\[y_k = \nabla f_{k+1} - \nabla f_k\]</div>
<div class="math">
\[s_k = x_{k+1} - x_k\]</div>
<p>Where <span class="math">\(f()\)</span> is the line search objective and <span class="math">\(x\)</span> the
vector of parameter values <a class="reference internal" href="bibliography.html#nocedalwright" id="id2">[NocedalWright]</a>.</p>
<p>It is important to note that approximate eigenvalue scaling does
<strong>not</strong> <em>always</em> improve convergence, and that it can in fact
<em>significantly</em> degrade performance for certain classes of problem,
which is why it is disabled by default. In particular it can
degrade performance when the sensitivity of the problem to different
parameters varies significantly, as in this case a single scalar
factor fails to capture this variation and detrimentally downscales
parts of the Jacobian approximation which correspond to
low-sensitivity parameters. It can also reduce the robustness of the
solution to errors in the Jacobians.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options30line_search_interpolation_typeE">
<span id="GradientProblemSolver::Options::line_search_interpolation_type__LineSearchIterpolationType"></span>LineSearchIterpolationType <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">line_search_interpolation_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options30line_search_interpolation_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">CUBIC</span></code></p>
<p>Degree of the polynomial used to approximate the objective
function. Valid values are <code class="docutils literal"><span class="pre">BISECTION</span></code>, <code class="docutils literal"><span class="pre">QUADRATIC</span></code> and
<code class="docutils literal"><span class="pre">CUBIC</span></code>.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options25min_line_search_step_sizeE">
<span id="GradientProblemSolver::Options::min_line_search_step_size__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">min_line_search_step_size</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options25min_line_search_step_sizeE" title="Permalink to this definition">¶</a></dt>
<dd><p>The line search terminates if:</p>
<div class="math">
\[\|\Delta x_k\|_\infty < \text{min_line_search_step_size}\]</div>
<p>where <span class="math">\(\|\cdot\|_\infty\)</span> refers to the max norm, and
<span class="math">\(\Delta x_k\)</span> is the step change in the parameter values at
the <span class="math">\(k\)</span>-th iteration.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options40line_search_sufficient_function_decreaseE">
<span id="GradientProblemSolver::Options::line_search_sufficient_function_decrease__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">line_search_sufficient_function_decrease</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options40line_search_sufficient_function_decreaseE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1e-4</span></code></p>
<p>Solving the line search problem exactly is computationally
prohibitive. Fortunately, line search based optimization algorithms
can still guarantee convergence if instead of an exact solution,
the line search algorithm returns a solution which decreases the
value of the objective function sufficiently. More precisely, we
are looking for a step size s.t.</p>
<div class="math">
\[f(\text{step_size}) \le f(0) + \text{sufficient_decrease} * [f'(0) * \text{step_size}]\]</div>
<p>This condition is known as the Armijo condition.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options32max_line_search_step_contractionE">
<span id="GradientProblemSolver::Options::max_line_search_step_contraction__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_line_search_step_contraction</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options32max_line_search_step_contractionE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1e-3</span></code></p>
<p>In each iteration of the line search,</p>
<div class="math">
\[\text{new_step_size} \geq \text{max_line_search_step_contraction} * \text{step_size}\]</div>
<p>Note that by definition, for contraction:</p>
<div class="math">
\[0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\]</div>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options32min_line_search_step_contractionE">
<span id="GradientProblemSolver::Options::min_line_search_step_contraction__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">min_line_search_step_contraction</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options32min_line_search_step_contractionE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">0.6</span></code></p>
<p>In each iteration of the line search,</p>
<div class="math">
\[\text{new_step_size} \leq \text{min_line_search_step_contraction} * \text{step_size}\]</div>
<p>Note that by definition, for contraction:</p>
<div class="math">
\[0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\]</div>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options40max_num_line_search_step_size_iterationsE">
<span id="GradientProblemSolver::Options::max_num_line_search_step_size_iterations__i"></span>int <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_num_line_search_step_size_iterations</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options40max_num_line_search_step_size_iterationsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">20</span></code></p>
<p>Maximum number of trial step size iterations during each line
search, if a step size satisfying the search conditions cannot be
found within this number of trials, the line search will stop.</p>
<p>As this is an ‘artificial’ constraint (one imposed by the user, not
the underlying math), if <code class="docutils literal"><span class="pre">WOLFE</span></code> line search is being used, <em>and</em>
points satisfying the Armijo sufficient (function) decrease
condition have been found during the current search (in <span class="math">\(\leq\)</span>
<code class="docutils literal"><span class="pre">max_num_line_search_step_size_iterations</span></code>). Then, the step size
with the lowest function value which satisfies the Armijo condition
will be returned as the new valid step, even though it does <em>not</em>
satisfy the strong Wolfe conditions. This behaviour protects
against early termination of the optimizer at a sub-optimal point.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options38max_num_line_search_direction_restartsE">
<span id="GradientProblemSolver::Options::max_num_line_search_direction_restarts__i"></span>int <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_num_line_search_direction_restarts</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options38max_num_line_search_direction_restartsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">5</span></code></p>
<p>Maximum number of restarts of the line search direction algorithm
before terminating the optimization. Restarts of the line search
direction algorithm occur when the current algorithm fails to
produce a new descent direction. This typically indicates a
numerical failure, or a breakdown in the validity of the
approximations used.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options41line_search_sufficient_curvature_decreaseE">
<span id="GradientProblemSolver::Options::line_search_sufficient_curvature_decrease__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">line_search_sufficient_curvature_decrease</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options41line_search_sufficient_curvature_decreaseE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">0.9</span></code></p>
<p>The strong Wolfe conditions consist of the Armijo sufficient
decrease condition, and an additional requirement that the
step size be chosen s.t. the <em>magnitude</em> (‘strong’ Wolfe
conditions) of the gradient along the search direction
decreases sufficiently. Precisely, this second condition
is that we seek a step size s.t.</p>
<div class="math">
\[\|f'(\text{step_size})\| \leq \text{sufficient_curvature_decrease} * \|f'(0)\|\]</div>
<p>Where <span class="math">\(f()\)</span> is the line search objective and <span class="math">\(f'()\)</span> is the derivative
of <span class="math">\(f\)</span> with respect to the step size: <span class="math">\(\frac{d f}{d~\text{step size}}\)</span>.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options30max_line_search_step_expansionE">
<span id="GradientProblemSolver::Options::max_line_search_step_expansion__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_line_search_step_expansion</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options30max_line_search_step_expansionE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">10.0</span></code></p>
<p>During the bracketing phase of a Wolfe line search, the step size
is increased until either a point satisfying the Wolfe conditions
is found, or an upper bound for a bracket containing a point
satisfying the conditions is found. Precisely, at each iteration
of the expansion:</p>
<div class="math">
\[\text{new_step_size} \leq \text{max_step_expansion} * \text{step_size}\]</div>
<p>By definition for expansion</p>
<div class="math">
\[\text{max_step_expansion} > 1.0\]</div>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options18max_num_iterationsE">
<span id="GradientProblemSolver::Options::max_num_iterations__i"></span>int <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_num_iterations</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options18max_num_iterationsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">50</span></code></p>
<p>Maximum number of iterations for which the solver should run.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options26max_solver_time_in_secondsE">
<span id="GradientProblemSolver::Options::max_solver_time_in_seconds__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">max_solver_time_in_seconds</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options26max_solver_time_in_secondsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1e6</span></code>
Maximum amount of time for which the solver should run.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options18function_toleranceE">
<span id="GradientProblemSolver::Options::function_tolerance__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">function_tolerance</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options18function_toleranceE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1e-6</span></code></p>
<p>Solver terminates if</p>
<div class="math">
\[\frac{|\Delta \text{cost}|}{\text{cost}} \leq \text{function_tolerance}\]</div>
<p>where, <span class="math">\(\Delta \text{cost}\)</span> is the change in objective
function value (up or down) in the current iteration of the line search.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options18gradient_toleranceE">
<span id="GradientProblemSolver::Options::gradient_tolerance__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">gradient_tolerance</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options18gradient_toleranceE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1e-10</span></code></p>
<p>Solver terminates if</p>
<div class="math">
\[\|x - \Pi \boxplus(x, -g(x))\|_\infty \leq \text{gradient_tolerance}\]</div>
<p>where <span class="math">\(\|\cdot\|_\infty\)</span> refers to the max norm, <span class="math">\(\Pi\)</span>
is projection onto the bounds constraints and <span class="math">\(\boxplus\)</span> is
Plus operation for the overall local parameterization associated
with the parameter vector.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options19parameter_toleranceE">
<span id="GradientProblemSolver::Options::parameter_tolerance__double"></span>double <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">parameter_tolerance</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options19parameter_toleranceE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">1e-8</span></code></p>
<p>Solver terminates if</p>
<div class="math">
\[\|\Delta x\| \leq (\|x\| + \text{parameter_tolerance}) * \text{parameter_tolerance}\]</div>
<p>where <span class="math">\(\Delta x\)</span> is the step computed by the linear solver in
the current iteration of the line search.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options12logging_typeE">
<span id="GradientProblemSolver::Options::logging_type__LoggingType"></span>LoggingType <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">logging_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options12logging_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">PER_MINIMIZER_ITERATION</span></code></p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options28minimizer_progress_to_stdoutE">
<span id="GradientProblemSolver::Options::minimizer_progress_to_stdout__b"></span>bool <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">minimizer_progress_to_stdout</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options28minimizer_progress_to_stdoutE" title="Permalink to this definition">¶</a></dt>
<dd><p>Default: <code class="docutils literal"><span class="pre">false</span></code></p>
<p>By default the <code class="xref cpp cpp-class docutils literal"><span class="pre">Minimizer</span></code> progress is logged to <code class="docutils literal"><span class="pre">STDERR</span></code>
depending on the <code class="docutils literal"><span class="pre">vlog</span></code> level. If this flag is set to true, and
<a class="reference internal" href="#_CPPv2N21GradientProblemSolver7Options12logging_typeE" title="GradientProblemSolver::Options::logging_type"><code class="xref cpp cpp-member docutils literal"><span class="pre">GradientProblemSolver::Options::logging_type</span></code></a> is not
<code class="docutils literal"><span class="pre">SILENT</span></code>, the logging output is sent to <code class="docutils literal"><span class="pre">STDOUT</span></code>.</p>
<p>The progress display looks like</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>0: f: 2.317806e+05 d: 0.00e+00 g: 3.19e-01 h: 0.00e+00 s: 0.00e+00 e: <span class="m">0</span> it: 2.98e-02 tt: 8.50e-02
1: f: 2.312019e+05 d: 5.79e+02 g: 3.18e-01 h: 2.41e+01 s: 1.00e+00 e: <span class="m">1</span> it: 4.54e-02 tt: 1.31e-01
2: f: 2.300462e+05 d: 1.16e+03 g: 3.17e-01 h: 4.90e+01 s: 2.54e-03 e: <span class="m">1</span> it: 4.96e-02 tt: 1.81e-01
</pre></div>
</div>
<p>Here</p>
<ol class="arabic simple">
<li><code class="docutils literal"><span class="pre">f</span></code> is the value of the objective function.</li>
<li><code class="docutils literal"><span class="pre">d</span></code> is the change in the value of the objective function if
the step computed in this iteration is accepted.</li>
<li><code class="docutils literal"><span class="pre">g</span></code> is the max norm of the gradient.</li>
<li><code class="docutils literal"><span class="pre">h</span></code> is the change in the parameter vector.</li>
<li><code class="docutils literal"><span class="pre">s</span></code> is the optimal step length computed by the line search.</li>
<li><code class="docutils literal"><span class="pre">it</span></code> is the time take by the current iteration.</li>
<li><code class="docutils literal"><span class="pre">tt</span></code> is the total time taken by the minimizer.</li>
</ol>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Options9callbacksE">
<span id="GradientProblemSolver::Options::callbacks__vector:IterationCallback:"></span>vector<IterationCallback> <code class="descclassname">GradientProblemSolver::Options::</code><code class="descname">callbacks</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Options9callbacksE" title="Permalink to this definition">¶</a></dt>
<dd><p>Callbacks that are executed at the end of each iteration of the
<code class="xref cpp cpp-class docutils literal"><span class="pre">Minimizer</span></code>. They are executed in the order that they are
specified in this vector. See the documentation for
<code class="xref cpp cpp-class docutils literal"><span class="pre">IterationCallback</span></code> for more details.</p>
<p>The solver does NOT take ownership of these pointers.</p>
</dd></dl>
</div>
<div class="section" id="gradientproblemsolver-summary">
<h3><a class="reference internal" href="#_CPPv2N21GradientProblemSolver7SummaryE" title="GradientProblemSolver::Summary"><code class="xref cpp cpp-class docutils literal"><span class="pre">GradientProblemSolver::Summary</span></code></a><a class="headerlink" href="#gradientproblemsolver-summary" title="Permalink to this headline">¶</a></h3>
<dl class="class">
<dt id="_CPPv2N21GradientProblemSolver7SummaryE">
<span id="GradientProblemSolver::Summary"></span><em class="property">class </em><code class="descclassname">GradientProblemSolver::</code><code class="descname">Summary</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7SummaryE" title="Permalink to this definition">¶</a></dt>
<dd><p>Summary of the various stages of the solver after termination.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK21GradientProblemSolver7Summary11BriefReportEv">
<span id="GradientProblemSolver::Summary::BriefReportC"></span>string <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">BriefReport</code><span class="sig-paren">(</span><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK21GradientProblemSolver7Summary11BriefReportEv" title="Permalink to this definition">¶</a></dt>
<dd><p>A brief one line description of the state of the solver after
termination.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK21GradientProblemSolver7Summary10FullReportEv">
<span id="GradientProblemSolver::Summary::FullReportC"></span>string <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">FullReport</code><span class="sig-paren">(</span><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK21GradientProblemSolver7Summary10FullReportEv" title="Permalink to this definition">¶</a></dt>
<dd><p>A full multiline description of the state of the solver after
termination.</p>
</dd></dl>
<dl class="function">
<dt id="_CPPv2NK21GradientProblemSolver7Summary16IsSolutionUsableEv">
<span id="GradientProblemSolver::Summary::IsSolutionUsableC"></span>bool <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">IsSolutionUsable</code><span class="sig-paren">(</span><span class="sig-paren">)</span> <em class="property">const</em><a class="headerlink" href="#_CPPv2NK21GradientProblemSolver7Summary16IsSolutionUsableEv" title="Permalink to this definition">¶</a></dt>
<dd><p>Whether the solution returned by the optimization algorithm can be
relied on to be numerically sane. This will be the case if
<cite>GradientProblemSolver::Summary:termination_type</cite> is set to <cite>CONVERGENCE</cite>,
<cite>USER_SUCCESS</cite> or <cite>NO_CONVERGENCE</cite>, i.e., either the solver
converged by meeting one of the convergence tolerances or because
the user indicated that it had converged or it ran to the maximum
number of iterations or time.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary16termination_typeE">
<span id="GradientProblemSolver::Summary::termination_type__TerminationType"></span>TerminationType <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">termination_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary16termination_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>The cause of the minimizer terminating.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary7messageE">
<span id="GradientProblemSolver::Summary::message__string"></span>string <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">message</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary7messageE" title="Permalink to this definition">¶</a></dt>
<dd><p>Reason why the solver terminated.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary12initial_costE">
<span id="GradientProblemSolver::Summary::initial_cost__double"></span>double <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">initial_cost</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary12initial_costE" title="Permalink to this definition">¶</a></dt>
<dd><p>Cost of the problem (value of the objective function) before the
optimization.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary10final_costE">
<span id="GradientProblemSolver::Summary::final_cost__double"></span>double <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">final_cost</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary10final_costE" title="Permalink to this definition">¶</a></dt>
<dd><p>Cost of the problem (value of the objective function) after the
optimization.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary10iterationsE">
<span id="GradientProblemSolver::Summary::iterations__vector:IterationSummary:"></span>vector<IterationSummary> <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">iterations</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary10iterationsE" title="Permalink to this definition">¶</a></dt>
<dd><p><code class="xref cpp cpp-class docutils literal"><span class="pre">IterationSummary</span></code> for each minimizer iteration in order.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv220num_cost_evaluations">
<span id="num_cost_evaluations__i"></span>int <code class="descclassname"></code><code class="descname">num_cost_evaluations</code><a class="headerlink" href="#_CPPv220num_cost_evaluations" title="Permalink to this definition">¶</a></dt>
<dd><p>Number of times the cost (and not the gradient) was evaluated.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv224num_gradient_evaluations">
<span id="num_gradient_evaluations__i"></span>int <code class="descclassname"></code><code class="descname">num_gradient_evaluations</code><a class="headerlink" href="#_CPPv224num_gradient_evaluations" title="Permalink to this definition">¶</a></dt>
<dd><p>Number of times the gradient (and the cost) were evaluated.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary21total_time_in_secondsE">
<span id="GradientProblemSolver::Summary::total_time_in_seconds__double"></span>double <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">total_time_in_seconds</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary21total_time_in_secondsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Time (in seconds) spent in the solver.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary31cost_evaluation_time_in_secondsE">
<span id="GradientProblemSolver::Summary::cost_evaluation_time_in_seconds__double"></span>double <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">cost_evaluation_time_in_seconds</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary31cost_evaluation_time_in_secondsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Time (in seconds) spent evaluating the cost vector.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary35gradient_evaluation_time_in_secondsE">
<span id="GradientProblemSolver::Summary::gradient_evaluation_time_in_seconds__double"></span>double <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">gradient_evaluation_time_in_seconds</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary35gradient_evaluation_time_in_secondsE" title="Permalink to this definition">¶</a></dt>
<dd><p>Time (in seconds) spent evaluating the gradient vector.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary14num_parametersE">
<span id="GradientProblemSolver::Summary::num_parameters__i"></span>int <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">num_parameters</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary14num_parametersE" title="Permalink to this definition">¶</a></dt>
<dd><p>Number of parameters in the problem.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary20num_local_parametersE">
<span id="GradientProblemSolver::Summary::num_local_parameters__i"></span>int <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">num_local_parameters</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary20num_local_parametersE" title="Permalink to this definition">¶</a></dt>
<dd><p>Dimension of the tangent space of the problem. This is different
from <a class="reference internal" href="#_CPPv2N21GradientProblemSolver7Summary14num_parametersE" title="GradientProblemSolver::Summary::num_parameters"><code class="xref cpp cpp-member docutils literal"><span class="pre">GradientProblemSolver::Summary::num_parameters</span></code></a> if a
<code class="xref cpp cpp-class docutils literal"><span class="pre">LocalParameterization</span></code> object is used.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary26line_search_direction_typeE">
<span id="GradientProblemSolver::Summary::line_search_direction_type__LineSearchDirectionType"></span>LineSearchDirectionType <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">line_search_direction_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary26line_search_direction_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Type of line search direction used.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary16line_search_typeE">
<span id="GradientProblemSolver::Summary::line_search_type__LineSearchType"></span>LineSearchType <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">line_search_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary16line_search_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>Type of the line search algorithm used.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary30line_search_interpolation_typeE">
<span id="GradientProblemSolver::Summary::line_search_interpolation_type__LineSearchInterpolationType"></span>LineSearchInterpolationType <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">line_search_interpolation_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary30line_search_interpolation_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>When performing line search, the degree of the polynomial used to
approximate the objective function.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary33nonlinear_conjugate_gradient_typeE">
<span id="GradientProblemSolver::Summary::nonlinear_conjugate_gradient_type__NonlinearConjugateGradientType"></span>NonlinearConjugateGradientType <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">nonlinear_conjugate_gradient_type</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary33nonlinear_conjugate_gradient_typeE" title="Permalink to this definition">¶</a></dt>
<dd><p>If the line search direction is <cite>NONLINEAR_CONJUGATE_GRADIENT</cite>,
then this indicates the particular variant of non-linear conjugate
gradient used.</p>
</dd></dl>
<dl class="member">
<dt id="_CPPv2N21GradientProblemSolver7Summary14max_lbfgs_rankE">
<span id="GradientProblemSolver::Summary::max_lbfgs_rank__i"></span>int <code class="descclassname">GradientProblemSolver::Summary::</code><code class="descname">max_lbfgs_rank</code><a class="headerlink" href="#_CPPv2N21GradientProblemSolver7Summary14max_lbfgs_rankE" title="Permalink to this definition">¶</a></dt>
<dd><p>If the type of the line search direction is <cite>LBFGS</cite>, then this
indicates the rank of the Hessian approximation.</p>
</dd></dl>
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