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<doc>
   <title>Optimization</title>

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   <body>
      <br/><br/>

         <p>
            This page documents library components that attempt to find the 
            minimum or maximum of a user supplied function.   An introduction
            to the general purpose non-linear optimizers in this section can be
            found <a href="optimization_ex.cpp.html">here</a>.  For an example
            showing how to use the non-linear least squares routines look 
            <a href="least_squares_ex.cpp.html">here</a>.
         </p>

   </body>

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   <menu width="150">
    <top>
      <section>
         <name>General Purpose Optimizers</name>
         <item>find_min</item> 
         <item>find_min_box_constrained</item> 
         <item>find_min_single_variable</item> 
         <item>find_min_using_approximate_derivatives</item> 
         <item>find_min_bobyqa</item> 
         <item>find_max</item> 
         <item>find_max_box_constrained</item> 
         <item>find_max_single_variable</item> 
         <item>find_max_using_approximate_derivatives</item> 
         <item>find_max_bobyqa</item> 
         <item>find_max_trust_region</item> 
         <item>find_min_trust_region</item> 
      </section>

      <section>
         <name>Special Purpose Optimizers</name>
         <item>solve_qp_using_smo</item> 
         <item>solve_qp2_using_smo</item> 
         <item>solve_qp3_using_smo</item> 
         <item>solve_qp4_using_smo</item> 
         <item>oca</item> 
         <item>solve_least_squares</item> 
         <item>solve_least_squares_lm</item> 
         <item>solve_trust_region_subproblem</item> 
         <item>max_cost_assignment</item> 
         <item>max_sum_submatrix</item> 
         <item>find_max_factor_graph_nmplp</item> 
         <item>find_max_factor_graph_viterbi</item> 
         <item>find_max_factor_graph_potts</item> 
         <item>find_max_parse_cky</item> 
         <item>min_cut</item> 
      </section>

      <section>
         <name>Strategies</name>
         <item>cg_search_strategy</item>
         <item>bfgs_search_strategy</item>
         <item>newton_search_strategy</item>
         <item>lbfgs_search_strategy</item>
         <item>objective_delta_stop_strategy</item>
         <item>gradient_norm_stop_strategy</item>
      </section>

      <section>
         <name>Helper Routines</name>
         <item>derivative</item> 
         <item>negate_function</item> 
         <item>clamp_function</item> 
         <item>make_line_search_function</item> 
         <item>poly_min_extrap</item> 
         <item>lagrange_poly_min_extrap</item> 
         <item>line_search</item> 
         <item>backtracking_line_search</item> 
         <item>graph_cut_score</item> 
         <item>potts_model_score</item> 
         <item>parse_tree_to_string</item> 
         <item>find_trees_not_rooted_with_tag</item> 
      </section>

    </top>  
   </menu>

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   <components>
   
   <!-- ************************************************************************* -->
      
      <component>
         <name>derivative</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
            This is a function that takes another function as input and returns
            a function object that numerically computes the derivative of the input function.
         </description>
                                 
      </component>


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      <component>
         <name>negate_function</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
            This is a function that takes another function as input and returns
            a function object that computes the negation of the input function.
         </description>
                                 
      </component>


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      <component>
         <name>clamp_function</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
            This is a function that takes another function, f(x), as input and 
            returns a new function object, g(x), such that  
            <tt>g(x) == f(clamp(x,x_lower,x_upper))</tt> where x_lower and x_upper 
            are vectors of box constraints which are applied to x.
         </description>
                                 
      </component>

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      <component>
         <name>make_line_search_function</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
            This is a function that takes another function f(x) as input and returns
            a function object l(z) = f(start + z*direction).   It is useful for
            turning multi-variable functions into single-variable functions for
            use with the <a href="#line_search">line_search</a> or 
            <a href="#backtracking_line_search">backtracking_line_search</a> routines.
         </description>
                                 
      </component>


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      <component>
         <name>poly_min_extrap</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
            This function finds the 2nd or 3rd degree polynomial that interpolates a 
            set of points and returns the minimum of that polynomial.
         </description>
                                 
      </component>

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      <component>
         <name>lagrange_poly_min_extrap</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
            This function finds the second order polynomial that interpolates a 
            set of points and returns the minimum of that polynomial.
         </description>
                                 
      </component>

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      <component>
         <name>line_search</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
            Performs a gradient based line search on a given function and returns the input
            that makes the function significantly smaller.  This implements the classic
            line search method using the strong Wolfe conditions with a bracketing and then
            sectioning phase, both using polynomial interpolation. 
         </description>
                                 
      </component>


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      <component>
         <name>backtracking_line_search</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
            Performs a line search on a given function and returns the input
            that makes the function significantly smaller.  This implementation uses a
            basic Armijo backtracking search with polynomial interpolation.
         </description>
                                 
      </component>

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      <component>
         <name>cg_search_strategy</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_search_strategies_abstract.h</spec_file>
         <description>
                This object represents a strategy for determining which direction
                a <a href="#line_search">line search</a> should be carried out along.  This particular object
                is an implementation of the Polak-Ribiere conjugate gradient method
                for determining this direction.

                  <p>
                This method uses an amount of memory that is linear in the number
                of variables to be optimized.  So it is capable of handling problems
                with a very large number of variables.  However, it is generally
                not as good as the L-BFGS algorithm (see the 
                <a href="#lbfgs_search_strategy">lbfgs_search_strategy</a> class).
                  </p>
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

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      <component>
         <name>bfgs_search_strategy</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_search_strategies_abstract.h</spec_file>
         <description>
                This object represents a strategy for determining which direction
                a <a href="#line_search">line search</a> should be carried out along.  This particular object
                is an implementation of the BFGS quasi-newton method for determining 
                this direction.

                  <p>
                This method uses an amount of memory that is quadratic in the number
                of variables to be optimized.  It is generally very effective but 
                if your problem has a very large number of variables then it isn't 
                appropriate.  Instead, you should try the <a href="#lbfgs_search_strategy">lbfgs_search_strategy</a>.
                  </p>
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

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      <component>
         <name>newton_search_strategy</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_search_strategies_abstract.h</spec_file>
         <description>
                This object represents a strategy for determining which direction
                a <a href="#line_search">line search</a> should be carried out along.  This particular routine 
                is an implementation of the newton method for determining this direction.  
                That means using it requires you to supply a method for
                creating hessian matrices for the problem you are trying to optimize.

               <p>
               Note also that this is actually a helper function for creating 
               <a href="dlib/optimization/optimization_search_strategies_abstract.h.html#newton_search_strategy_obj"
                     >newton_search_strategy_obj</a> objects.  
               </p>

         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

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      <component>
         <name>lbfgs_search_strategy</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_search_strategies_abstract.h</spec_file>
         <description>
                This object represents a strategy for determining which direction
                a <a href="#line_search">line search</a> should be carried out along.  This particular object
                is an implementation of the L-BFGS quasi-newton method for determining 
                this direction.

                  <p>
                This method uses an amount of memory that is linear in the number
                of variables to be optimized.  This makes it an excellent method 
                to use when an optimization problem has a large number of variables.
                  </p>
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>objective_delta_stop_strategy</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_stop_strategies_abstract.h</spec_file>
         <description>
                This object represents a strategy for deciding if an optimization
                algorithm should terminate.   This particular object looks at the 
                change in the objective function from one iteration to the next and 
                bases its decision on how large this change is.  If the change
                is below a user given threshold then the search stops.
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>gradient_norm_stop_strategy</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_stop_strategies_abstract.h</spec_file>
         <description>
                This object represents a strategy for deciding if an optimization
                algorithm should terminate.   This particular object looks at the 
                norm (i.e. the length) of the current gradient vector and
                stops if it is smaller than a user given threshold.  
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_min</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
             Performs an unconstrained minimization of a nonlinear function using 
             some search strategy (e.g. <a href="#bfgs_search_strategy">bfgs_search_strategy</a>).
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>find_min_box_constrained</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
             Performs a box constrained minimization of a nonlinear function using 
             some search strategy (e.g. <a href="#bfgs_search_strategy">bfgs_search_strategy</a>).
             This function uses a backtracking line search along with a gradient projection
             step to handle the box constraints.
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>find_min_single_variable</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
             Performs a bound constrained minimization of a nonlinear function.  The 
             function must be of a single variable.  Derivatives are not required.  
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>solve_trust_region_subproblem</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_trust_region_abstract.h</spec_file>
         <description>
             This function solves the following optimization problem:
<pre>
Minimize: f(p) == 0.5*trans(p)*B*p + trans(g)*p
subject to the following constraint:
   length(p) &lt;= radius
</pre>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>solve_qp_using_smo</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_solve_qp_using_smo_abstract.h</spec_file>
         <description>
             This function solves the following quadratic program:
<pre>
   Minimize: f(alpha) == 0.5*trans(alpha)*Q*alpha - trans(alpha)*b
   subject to the following constraints:
      sum(alpha) == C 
      min(alpha) >= 0 
   Where f is convex.  This means that Q should be symmetric and positive-semidefinite.
</pre>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>solve_qp2_using_smo</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_solve_qp2_using_smo_abstract.h</spec_file>
         <description>
             This function solves the following quadratic program:
<pre>
   Minimize: f(alpha) == 0.5*trans(alpha)*Q*alpha 
   subject to the following constraints:
      sum(alpha) == nu*y.size() 
      0 &lt;= min(alpha) &amp;&amp; max(alpha) &lt;= 1 
      trans(y)*alpha == 0

   Where all elements of y must be equal to +1 or -1 and f is convex.  
   This means that Q should be symmetric and positive-semidefinite.
</pre>
<br/>
                This object implements the strategy used by the LIBSVM tool.  The following papers
                can be consulted for additional details:
               <ul>
                  <li>Chang and Lin, Training {nu}-Support Vector Classifiers: Theory and Algorithms</li>
                  <li>Chih-Chung Chang and Chih-Jen Lin, LIBSVM : a library for support vector 
                     machines, 2001. Software available at 
                     <a href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/">http://www.csie.ntu.edu.tw/~cjlin/libsvm</a></li>
               </ul>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>solve_qp3_using_smo</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_solve_qp3_using_smo_abstract.h</spec_file>
         <description>
             This function solves the following quadratic program:
<pre>
   Minimize: f(alpha) == 0.5*trans(alpha)*Q*alpha + trans(p)*alpha
   subject to the following constraints:
        for all i such that y(i) == +1:  0 &lt;= alpha(i) &lt;= Cp 
        for all i such that y(i) == -1:  0 &lt;= alpha(i) &lt;= Cn 
        trans(y)*alpha == B 

   Where all elements of y must be equal to +1 or -1 and f is convex.  
   This means that Q should be symmetric and positive-semidefinite.
</pre>
<br/>
                This object implements the strategy used by the LIBSVM tool.  The following papers
                can be consulted for additional details:
               <ul>
                  <li>Chih-Chung Chang and Chih-Jen Lin, LIBSVM : a library for support vector 
                     machines, 2001. Software available at 
                     <a href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/">http://www.csie.ntu.edu.tw/~cjlin/libsvm</a></li>
                  <li>Working Set Selection Using Second Order Information for Training Support Vector Machines by
                     Fan, Chen, and Lin.  In the Journal of Machine Learning Research 2005.</li>
               </ul>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>solve_qp4_using_smo</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_solve_qp_using_smo_abstract.h</spec_file>
         <description>
             This function solves the following quadratic program:
<pre>
   Minimize: f(alpha,lambda) == 0.5*trans(alpha)*Q*alpha - trans(alpha)*b + 
                                0.5*trans(lambda)*lambda - trans(lambda)*A*alpha
   subject to the following constraints:
      sum(alpha)  == C 
      min(alpha)  >= 0 
      min(lambda) >= 0
   Where f is convex.  This means that Q should be positive-semidefinite.
</pre>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>max_cost_assignment</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/max_cost_assignment_abstract.h</spec_file>
         <description>
            This function is an implementation of the Hungarian algorithm (also know as the Kuhn-Munkres algorithm) which
            runs in O(N^3) time.
            It solves the optimal assignment problem. For example, suppose you have an equal number of workers
            and jobs and you need to decide which workers to assign to which jobs. Some workers are better at 
            certain jobs than others. So you would like to find out how to assign them all to jobs such that 
            overall productivity is maximized. You can use this routine to solve this problem and others like it. 
            <p>
               Note that dlib also contains a <a href="ml.html#structural_assignment_trainer">machine learning</a>
               method for learning the cost function needed to use the Hungarian algorithm.
            </p>

         </description>
         <examples>
            <example>max_cost_assignment.py.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>max_sum_submatrix</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/max_sum_submatrix_abstract.h</spec_file>
         <description>
              This function finds the submatrix within a user supplied matrix which has the largest sum.  It then
              zeros out that submatrix and repeats the process until no more maximal submatrices can 
              be found.   
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_factor_graph_nmplp</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/find_max_factor_graph_nmplp_abstract.h</spec_file>
         <description>
              This function is a tool for approximately solving the MAP problem in a graphical 
              model or factor graph with pairwise potential functions.  That is, it attempts 
              to solve a certain kind of optimization problem which can be defined as follows:
<pre>
   maximize: f(X)
   where X is a set of integer valued variables and f(X) can be written
   as the sum of functions which each involve only two variables from X.
</pre>
If the graph is tree-structured then this routine always gives the exact solution to the MAP problem.
However, for graphs with cycles, the solution may be approximate.
<br/>
<br/>
            This function is an implementation of the NMPLP algorithm introduced in the 
            following papers:
                <blockquote>
                Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations (2008)
                by Amir Globerson and Tommi Jaakkola
                </blockquote>
                <blockquote>
                Introduction to dual decomposition for inference (2011)
                by David Sontag, Amir Globerson, and Tommi Jaakkola 
                </blockquote>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_parse_cky</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/find_max_parse_cky_abstract.h</spec_file>
         <description>
            This function implements the CKY parsing algorithm.  In particular, it
            finds the maximum scoring binary parse tree that parses an input sequence of tokens.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>parse_tree_to_string</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/find_max_parse_cky_abstract.h</spec_file>
         <description>
            This is a set of functions useful for converting a parse tree output by 
            <a href="#find_max_parse_cky">find_max_parse_cky</a> into a bracketed string
            suitable for displaying the parse tree.  
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>find_trees_not_rooted_with_tag</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/find_max_parse_cky_abstract.h</spec_file>
         <description>
              Finds all the largest non-overlapping <a href="#find_max_parse_cky">parse trees</a> 
              in tree that are not rooted with a particular tag.   
              <p>
              This function is useful when you want to cut a parse tree
              into a bunch of sub-trees and you know that the top level of the tree is all
              composed of the same kind of tag.  So if you want to just "slice off" the top
              of the tree where this tag lives then this function is useful for doing that.
              </p>
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>find_max_factor_graph_viterbi</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/find_max_factor_graph_viterbi_abstract.h</spec_file>
         <description>
            This function is a tool for exactly solving the MAP problem in a chain-structured 
            graphical model or factor graph.   In particular, it is an implementation of the classic Viterbi
            algorithm for finding the maximizing assignment.  In addition to basic first order Markov
            models, this function is also capable of finding the MAP assignment for higher order
            Markov models.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>potts_model_score</name>
         <file>dlib/graph_cuts.h</file>
         <spec_file link="true">dlib/graph_cuts/find_max_factor_graph_potts_abstract.h</spec_file>
         <description>
              This routine computes the model score for a Potts problem and a
              candidate labeling.  This score is the quantity maximised
              by the <a href="#find_max_factor_graph_potts">find_max_factor_graph_potts</a>
              routine.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>graph_cut_score</name>
         <file>dlib/graph_cuts.h</file>
         <spec_file link="true">dlib/graph_cuts/min_cut_abstract.h</spec_file>
         <description>
            This routine computes the score for a candidate graph cut.  This is the 
            quantity minimized by the <a href="#min_cut">min_cut</a> algorithm.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>min_cut</name>
         <file>dlib/graph_cuts.h</file>
         <spec_file link="true">dlib/graph_cuts/min_cut_abstract.h</spec_file>
         <description>
                This is a function object which can be used to find the min cut
                on a graph.
                The implementation is based on the method described in the following
                paper:
                <blockquote>
                    An Experimental Comparison of Min-Cut/Max-Flow Algorithms for
                    Energy Minimization in Vision, by Yuri Boykov and Vladimir Kolmogorov, 
                    in PAMI 2004.
                </blockquote>
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_factor_graph_potts</name>
         <file>dlib/graph_cuts.h</file>
         <spec_file link="true">dlib/graph_cuts/find_max_factor_graph_potts_abstract.h</spec_file>
         <description>
              This is a set of overloaded functions for exactly solving the MAP problem in a Potts
              model.  This type of model is useful when you have a problem which
              can be modeled as a bunch of binary decisions on some variables,
              but you have some kind of labeling consistency constraint.  This
              means that there is some penalty for giving certain pairs of variables
              different labels.  So in addition to trying to figure out how to best
              label each variable on its own, you have to worry about making the
              labels pairwise consistent in some sense.  The find_max_factor_graph_potts()
              routine can be used to find the most probable/highest scoring
              labeling for this type of model.
              <p>The implementation of this routine is based on the <a href="#min_cut">min_cut</a> object.</p>
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>oca</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_oca_abstract.h</spec_file>
         <description>
                This object is a tool for solving the following optimization problem:   
<pre>
   Minimize: f(w) == 0.5*dot(w,w) + C*R(w)

   Where R(w) is a user-supplied convex function and C > 0.  Optionally,
   this object can also add non-negativity constraints to some or all
   of the elements of w.
</pre>
<br/>
<br/>

                For a detailed discussion you should consult the following papers
                from the Journal of Machine Learning Research:
                <blockquote>
                    Optimized Cutting Plane Algorithm for Large-Scale Risk Minimization
                      by  Vojtech Franc, Soren Sonnenburg; 10(Oct):2157--2192, 2009. 
                </blockquote>
                <blockquote>
                    Bundle Methods for Regularized Risk Minimization
                      by Choon Hui Teo, S.V.N. Vishwanthan, Alex J. Smola, Quoc V. Le; 11(Jan):311-365, 2010. 
                </blockquote>

         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      
      <component>
         <name>find_min_bobyqa</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_bobyqa_abstract.h</spec_file>
         <description>
            This function defines the dlib interface to the BOBYQA software developed by M.J.D Powell.
            BOBYQA is a method for optimizing a function in the absence of derivative information.  
            Powell described it as a method that seeks the least value of a function of many 
            variables, by applying a trust region method that forms quadratic models by 
            interpolation.  There is usually some freedom in the interpolation conditions, 
            which is taken up by minimizing the Frobenius norm of the change to the second 
            derivative of the model, beginning with the zero matrix. The values of the variables 
            are constrained by upper and lower bounds.  

            <p>
            The following paper, published in 2009 by Powell, describes the
            detailed working of the BOBYQA algorithm.  

               <blockquote>
               The BOBYQA algorithm for bound constrained optimization 
               without derivatives by M.J.D. Powell
               </blockquote>
            </p>

            <p>
               Note that BOBYQA only works on functions of two or more variables.  So if you need to perform 
               derivative-free optimization on a function of a single variable 
               then you should use the <a href="#find_min_single_variable">find_min_single_variable</a>
               function.
            </p>

         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
            <example>model_selection_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_bobyqa</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_bobyqa_abstract.h</spec_file>
         <description>
            This function is identical to the <a href="#find_min_bobyqa">find_min_bobyqa</a> routine 
            except that it negates the objective function before performing optimization.  
            Thus this function will attempt to find the maximizer of the objective rather than 
            the minimizer.
            <p>
               Note that BOBYQA only works on functions of two or more variables.  So if you need to perform 
               derivative-free optimization on a function of a single variable 
               then you should use the <a href="#find_max_single_variable">find_max_single_variable</a>
               function.
            </p>
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
            <example>model_selection_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_min_using_approximate_derivatives</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
             Performs an unconstrained minimization of a nonlinear function using 
             some search strategy (e.g. <a href="#bfgs_search_strategy">bfgs_search_strategy</a>).
             This version doesn't take a gradient function but instead numerically approximates 
             the gradient.
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->

      <component>
         <name>solve_least_squares</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_least_squares_abstract.h</spec_file>
         <description>
            This is a function for solving non-linear least squares problems.  It uses a method
            which combines the traditional Levenberg-Marquardt technique with a quasi-newton
            approach.  It is appropriate for large residual problems (i.e. problems where the
            terms in the least squares function, the residuals, don't go to zero but remain
            large at the solution)
         </description>
                                 
         <examples>
            <example>least_squares_ex.cpp.html</example>
         </examples>
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>solve_least_squares_lm</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_least_squares_abstract.h</spec_file>
         <description>
            This is a function for solving non-linear least squares problems.  It uses 
            the traditional Levenberg-Marquardt technique. 
            It is appropriate for small residual problems (i.e. problems where the
            terms in the least squares function, the residuals, go to zero at the solution)
         </description>
                                 
         <examples>
            <example>least_squares_ex.cpp.html</example>
         </examples>
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_min_trust_region</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_trust_region_abstract.h</spec_file>
         <description>
             Performs an unconstrained minimization of a nonlinear function using 
             a trust region method.
         </description>
         <examples>
            <example>optimization_ex.cpp.html</example>
         </examples>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_trust_region</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_trust_region_abstract.h</spec_file>
         <description>
             Performs an unconstrained maximization of a nonlinear function using 
             a trust region method.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
             Performs an unconstrained maximization of a nonlinear function using 
             some search strategy (e.g. <a href="#bfgs_search_strategy">bfgs_search_strategy</a>).
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_box_constrained</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
             Performs a box constrained maximization of a nonlinear function using 
             some search strategy (e.g. <a href="#bfgs_search_strategy">bfgs_search_strategy</a>).
             This function uses a backtracking line search along with a gradient projection
             step to handle the box constraints.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_single_variable</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_line_search_abstract.h</spec_file>
         <description>
             Performs a bound constrained maximization of a nonlinear function.  The 
             function must be of a single variable.  Derivatives are not required.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->
      
      <component>
         <name>find_max_using_approximate_derivatives</name>
         <file>dlib/optimization.h</file>
         <spec_file link="true">dlib/optimization/optimization_abstract.h</spec_file>
         <description>
             Performs an unconstrained maximization of a nonlinear function using 
             some search strategy (e.g. <a href="#bfgs_search_strategy">bfgs_search_strategy</a>).
             This version doesn't take a gradient function but instead numerically approximates 
             the gradient.
         </description>
                                 
      </component>

   <!-- ************************************************************************* -->


   </components>

   <!-- ************************************************************************* -->


</doc>

