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|
<?xml version="1.0" encoding="utf8"?>
<?xml-stylesheet type="text/xsl" href="stylesheet.xsl"?>
<doc>
<title>Optimization</title>
<!-- ************************************************************************* -->
<body>
<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>
<!-- ************************************************************************* -->
<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_min_global</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_global</item>
<item>global_function_search</item>
<item>find_max_trust_region</item>
<item>find_min_trust_region</item>
</section>
<section>
<name>Special Purpose Optimizers</name>
<item>find_gap_between_convex_hulls</item>
<item>solve_qp_box_constrained</item>
<item>solve_qp_box_constrained_blockdiag</item>
<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>mpc</item>
<item>solve_least_squares</item>
<item>solve_least_squares_lm</item>
<item>solve_trust_region_subproblem</item>
<item>solve_trust_region_subproblem_bounded</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>
<item>elastic_net</item>
<item>isotonic_regression</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>
<item>upper_bound_function</item>
<item>call_function_and_expand_args</item>
</section>
</top>
</menu>
<!-- ************************************************************************* -->
<!-- ************************************************************************* -->
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<component>
<name>negate_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 as input and returns
a function object that computes the negation of the input function.
</description>
</component>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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>
<!-- ************************************************************************* -->
<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_bounded</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) <= radius
lower(i) <= p(i) <= upper(i), for all i
</pre>
</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) <= radius
</pre>
</description>
</component>
<!-- ************************************************************************* -->
<component>
<name>find_gap_between_convex_hulls</name>
<file>dlib/optimization.h</file>
<spec_file link="true">dlib/optimization/optimization_solve_qp_using_smo_abstract.h</spec_file>
<description>
This function measures the position and size of the gap between two convex
polytopes. In particular, it solves the following quadratic program:
<pre>
Minimize: f(cA,cB) == length_squared(A*cA - B*cB)
subject to the following constraints on cA and cB:
- is_col_vector(cA) == true && cA.size() == A.nc()
- is_col_vector(cB) == true && cB.size() == B.nc()
- sum(cA) == 1 && min(cA) >= 0
- sum(cB) == 1 && min(cB) >= 0
</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_qp_box_constrained</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(b)*alpha
subject to the following box constraints on alpha:
0 <= min(alpha-lower)
0 <= max(upper-alpha)
Where f is convex. This means that Q should be positive-semidefinite.
</pre>
</description>
</component>
<!-- ************************************************************************* -->
<component>
<name>solve_qp_box_constrained_blockdiag</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(b)*alpha
subject to the following box constraints on alpha:
0 <= min(alpha-lower)
0 <= max(upper-alpha)
Where f is convex. This means that Q should be positive-semidefinite.
</pre>
So it does the same thing as <a href="#solve_qp_box_constrained">solve_qp_box_constrained</a>,
except it is optimized for large Q matrices with a special block
structure. In particular, Q must be grouped into identically sized
blocks where all blocks are diagonal matrices, except those on the
main diagonal which can be dense.
</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 <= min(alpha) && max(alpha) <= 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 <= alpha(i) <= Cp
for all i such that y(i) == -1: 0 <= alpha(i) <= 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 - trans(lambda)*d
subject to the following constraints:
sum(alpha) == C
min(alpha) >= 0
min(lambda) >= 0
max(lambda) <= max_lambda
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_ex.cpp.html</example>
<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*||w||^2 + 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.
Or it can alternatively solve:
Minimize: f(w) == 0.5*||w-prior||^2 + C*R(w)
Where prior is a user supplied vector and R(w) has the same
interpretation as above.
Or it can use the elastic net regularizer:
Minimize: f(w) == 0.5*(1-lasso_lambda)*length_squared(w) + lasso_lambda*sum(abs(w)) + C*R(w)
Where lasso_lambda is a number in the range [0, 1) and controls
trade-off between doing L1 and L2 regularization. R(w) has the same
interpretation as above.
</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>mpc</name>
<file>dlib/control.h</file>
<spec_file link="true">dlib/control/mpc_abstract.h</spec_file>
<description>
This object implements a linear model predictive controller.
In particular, it solves a certain quadratic program using the method
described in the paper:
<blockquote>
A Fast Gradient method for embedded linear predictive control (2011)
by Markus Kogel and Rolf Findeisen
</blockquote>
</description>
<examples>
<example>mpc_ex.cpp.html</example>
</examples>
</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>
</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>
</examples>
</component>
<!-- ************************************************************************* -->
<component>
<name>isotonic_regression</name>
<file>dlib/optimization.h</file>
<spec_file link="true">dlib/optimization/isotonic_regression_abstract.h</spec_file>
<description>
This object is a tool for performing 1-D isotonic regression. That is, it
finds the least squares fit of a non-parametric curve to some user supplied
data, subject to the constraint that the fitted curve is non-decreasing.
<p>
This is done using the fast O(n) pool adjacent violators algorithm.
</p>
</description>
</component>
<!-- ************************************************************************* -->
<component>
<name>elastic_net</name>
<file>dlib/optimization/elastic_net.h</file>
<spec_file link="true">dlib/optimization/elastic_net_abstract.h</spec_file>
<description>
This object is a tool for solving the following optimization problem:
<pre>
min_w: length_squared(X*w - Y) + ridge_lambda*length_squared(w)
such that: sum(abs(w)) <= lasso_budget
</pre>
<p>
That is, it solves the elastic net optimization problem. This object also
has the special property that you can quickly obtain different solutions
for different settings of ridge_lambda, lasso_budget, and target Y values.
</p>
<p>
This is because a large amount of work is precomputed in the constructor.
The solver will also remember the previous solution and will use that to
warm start subsequent invocations. Therefore, you can efficiently get
solutions for a wide range of regularization parameters.
</p>
The particular algorithm used to solve it is described in the paper:
<blockquote>
Zhou, Quan, et al. "A reduction of the elastic net to support vector
machines with an application to gpu computing." arXiv preprint
arXiv:1409.1976 (2014). APA
</blockquote>
And for the SVM solver sub-component we use the algorithm from:
<blockquote>
Hsieh, Cho-Jui, et al. "A dual coordinate descent method for large-scale
linear SVM." Proceedings of the 25th international conference on Machine
learning. ACM, 2008.
</blockquote>
</description>
</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>
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<component>
<name>global_function_search</name>
<file>dlib/global_optimization.h</file>
<spec_file link="true">dlib/global_optimization/global_function_search_abstract.h</spec_file>
<description>
This object performs global optimization of a set of user supplied
functions. That is, given a set of functions, each of which could take a different
number of arguments, this object allows you to find which function and which arguments
produce the maximal output.
<p>
Importantly, the global_function_search object does not require the user to
supply derivatives. Moreover, the functions being optimized may contain discontinuities,
behave stochastically, and have many local maxima. The global_function_search
object will attempt to find the global optima in the face of these challenges.
It is also designed to use as few function evaluations as possible, making
it suitable for optimizing functions that are very expensive to evaluate.
It does this by alternating between two modes: a global exploration mode
and a local optima refinement mode. This is accomplished by building and
maintaining two models of the objective function:
</p>
<ol>
<li>
A global model that <a href="#upper_bound_function">upper bounds our objective function</a>. This is a non-parametric
piecewise linear model derived from all function evaluations ever seen by the
global_function_search object. This is based on the method described in <i>Global
Optimization of Lipschitz Functions</i> by Cédric Malherbe and Nicolas Vayatis in the
2017 International Conference on Machine Learning.
</li>
<li>
A local quadratic model fit around the best point seen so far. This uses
a trust region method similar to what is proposed in:
<i>The NEWUOA software for unconstrained optimization without derivatives</i> By
M.J.D. Powell, 40th Workshop on Large Scale Nonlinear Optimization (Erice,
Italy, 2004)
</li>
</ol>
The behavior of the algorithm is illustrated in the following video, which shows the solver in action. In the video, the red line
is the function to be optimized and we are looking for the maximum point. Every time
the global_function_search samples a point from the function we note it with a little
box. The state of the solver is determined by the two models discussed above.
Therefore, we draw the upper bounding model as well as the current local quadratic model
so you can see how they evolve as the optimization proceeds. We also note the location of the
best point seen so far by a little vertical line.
<p>
You can see that the optimizer is alternating between picking the maximum upper bounding
point and the maximum point according to the quadratic model. As the optimization
progresses, the upper bound becomes progressively more accurate, helping to find the
best peak to investigate, while the quadratic model quickly finds a high precision
maximizer on whatever peak it currently rests. These two things together allow the
optimizer to find the true global maximizer to high precision (within 1e-9 in this case) by the time the
video concludes.
</p>
<center>
<video src="find_max_global_example">
Video of optimizer running
</video>
</center>
<p>
Finally, note that the <a href="#find_max_global">find_max_global</a> routine is
essentially a simple wrapper around the global_function_search object and exists to
provide a convenient interface. Most users will therefore want to call find_max_global
rather than global_function_search. However, the API of global_function_search
is more general and allows for of a wider set of usage patterns, for example, executing
objective function evaluations in parallel. So more advanced users may want to use
global_function_search directly rather than find_max_global. But try to use find_max_global() first.
</p>
</description>
</component>
<!-- ************************************************************************* -->
<component>
<name>find_max_global</name>
<file>dlib/global_optimization.h</file>
<spec_file link="true">dlib/global_optimization/find_max_global_abstract.h</spec_file>
<description>
This function performs global optimization of a function, subject
to bounds constraints. This means it attempts to find the global
maximizer, not just a local maximizer. The search is performed
using the <a href="#global_function_search">global_function_search</a> object.
See global_function_search's documentation for details of the algorithm. Importantly,
find_max_global() does not require the user to specify derivatives
or starting guesses, all while attempting to use as few calls to
the objective function as possible. It is therefore appropriate for tasks
where evaluating the objective function is time consuming or
expensive, such as in hyper parameter optimization of machine
learning models.
</description>
<examples>
<example>optimization_ex.cpp.html</example>
<example>model_selection_ex.cpp.html</example>
<example>global_optimization.py.html</example>
</examples>
</component>
<!-- ************************************************************************* -->
<component>
<name>find_min_global</name>
<file>dlib/global_optimization.h</file>
<spec_file link="true">dlib/global_optimization/find_max_global_abstract.h</spec_file>
<description>
This function is identical to the <a href="#find_max_global">find_max_global</a> routine
except it negates the objective function before performing optimization.
Thus this function will attempt to find the minimizer of the objective rather than
the maximizer.
</description>
<examples>
<example>optimization_ex.cpp.html</example>
<example>model_selection_ex.cpp.html</example>
<example>global_optimization.py.html</example>
</examples>
</component>
<!-- ************************************************************************* -->
<component>
<name>upper_bound_function</name>
<file>dlib/global_optimization.h</file>
<spec_file link="true">dlib/global_optimization/upper_bound_function_abstract.h</spec_file>
<description>
This object represents a piecewise linear non-parametric function that can
be used to define an upper bound on some more complex and unknown function.
<p>
This is based on the method described in <i>Global Optimization of Lipschitz
Functions</i> by Cédric Malherbe and Nicolas Vayatis in the 2017 International
Conference on Machine Learning. Here we have extended it to support modeling of
stochastic or discontinuous functions by adding a noise term. We also model separate
Lipschitz parameters for each dimension, allowing the model to handle functions with
widely varying sensitivities to each input variable.
</p>
</description>
</component>
<!-- ************************************************************************* -->
<component>
<name>call_function_and_expand_args</name>
<file>dlib/global_optimization.h</file>
<spec_file link="true">dlib/global_optimization/find_max_global_abstract.h</spec_file>
<description>
This routine allows you to pass a <tt>dlib::matrix<double,0,1></tt> object to
a function that takes simple <tt>double</tt> arguments. It does this by automatically
expanding the matrix elements and invoking the function. For example, suppose you had
a function like this:
<code_box>
double f(double x, double y, double z); </code_box>
You could then call f() like this:
<code_box>
matrix<double,0,1> args = {3,4,5};
call_function_and_expand_args(f, args); // calls: f(3,4,5) </code_box>
This kind of thing is convenient when writing optimizers like <a
href="#find_max_global">find_max_global</a> since it allows a wide range of
input functions to be given to the optimizer, including functions with
explicitly named arguments like x,y,z as shown above.
</description>
</component>
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</components>
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</doc>
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