# Optimization

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 here. For an example showing how to use the non-linear least squares routines look here.

General Purpose Optimizers
find_min
find_min_box_constrained
find_min_single_variable
find_min_using_approximate_derivatives
find_min_bobyqa
find_max
find_max_box_constrained
find_max_single_variable
find_max_using_approximate_derivatives
find_max_bobyqa
find_max_trust_region
find_min_trust_region
Special Purpose Optimizers
solve_qp_using_smo
solve_qp2_using_smo
solve_qp3_using_smo
solve_qp4_using_smo
oca
solve_least_squares
solve_least_squares_lm
solve_trust_region_subproblem
max_cost_assignment
max_sum_submatrix
find_max_factor_graph_nmplp
find_max_factor_graph_viterbi
find_max_factor_graph_potts
find_max_parse_cky
min_cut
Strategies
cg_search_strategy
bfgs_search_strategy
newton_search_strategy
lbfgs_search_strategy
objective_delta_stop_strategy
gradient_norm_stop_strategy
Helper Routines
derivative
negate_function
clamp_function
make_line_search_function
poly_min_extrap
lagrange_poly_min_extrap
line_search
backtracking_line_search
graph_cut_score
potts_model_score
parse_tree_to_string
find_trees_not_rooted_with_tag
derivative
dlib/optimization.h
dlib/optimization/optimization_abstract.h
This is a function that takes another function as input and returns a function object that numerically computes the derivative of the input function.
negate_function
dlib/optimization.h
dlib/optimization/optimization_abstract.h
This is a function that takes another function as input and returns a function object that computes the negation of the input function.
clamp_function
dlib/optimization.h
dlib/optimization/optimization_abstract.h
This is a function that takes another function, f(x), as input and returns a new function object, g(x), such that g(x) == f(clamp(x,x_lower,x_upper)) where x_lower and x_upper are vectors of box constraints which are applied to x.
make_line_search_function
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
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 line_search or backtracking_line_search routines.
poly_min_extrap
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
This function finds the 2nd or 3rd degree polynomial that interpolates a set of points and returns the minimum of that polynomial.
lagrange_poly_min_extrap
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
This function finds the second order polynomial that interpolates a set of points and returns the minimum of that polynomial.
line_search
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
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.
backtracking_line_search
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
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.
cg_search_strategy
dlib/optimization.h
dlib/optimization/optimization_search_strategies_abstract.h
This object represents a strategy for determining which direction a line search should be carried out along. This particular object is an implementation of the Polak-Ribiere conjugate gradient method for determining this direction.

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 lbfgs_search_strategy class).

optimization_ex.cpp.html
bfgs_search_strategy
dlib/optimization.h
dlib/optimization/optimization_search_strategies_abstract.h
This object represents a strategy for determining which direction a line search should be carried out along. This particular object is an implementation of the BFGS quasi-newton method for determining this direction.

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 lbfgs_search_strategy.

optimization_ex.cpp.html
newton_search_strategy
dlib/optimization.h
dlib/optimization/optimization_search_strategies_abstract.h
This object represents a strategy for determining which direction a line search 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.

Note also that this is actually a helper function for creating newton_search_strategy_obj objects.

optimization_ex.cpp.html
lbfgs_search_strategy
dlib/optimization.h
dlib/optimization/optimization_search_strategies_abstract.h
This object represents a strategy for determining which direction a line search should be carried out along. This particular object is an implementation of the L-BFGS quasi-newton method for determining this direction.

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.

optimization_ex.cpp.html
objective_delta_stop_strategy
dlib/optimization.h
dlib/optimization/optimization_stop_strategies_abstract.h
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.
optimization_ex.cpp.html
gradient_norm_stop_strategy
dlib/optimization.h
dlib/optimization/optimization_stop_strategies_abstract.h
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.
find_min
dlib/optimization.h
dlib/optimization/optimization_abstract.h
Performs an unconstrained minimization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy).
optimization_ex.cpp.html
find_min_box_constrained
dlib/optimization.h
dlib/optimization/optimization_abstract.h
Performs a box constrained minimization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy). This function uses a backtracking line search along with a gradient projection step to handle the box constraints.
optimization_ex.cpp.html
find_min_single_variable
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
Performs a bound constrained minimization of a nonlinear function. The function must be of a single variable. Derivatives are not required.
solve_trust_region_subproblem
dlib/optimization.h
dlib/optimization/optimization_trust_region_abstract.h
This function solves the following optimization problem:

    Minimize: f(p) == 0.5*trans(p)*B*p + trans(g)*p
    subject to the following constraint:
       length(p) <= radius

solve_qp_using_smo
dlib/optimization.h
dlib/optimization/optimization_solve_qp_using_smo_abstract.h
This function solves the following quadratic program:

       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.

solve_qp2_using_smo
dlib/optimization.h
dlib/optimization/optimization_solve_qp2_using_smo_abstract.h
This function solves the following quadratic program:

       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.

This object implements the strategy used by the LIBSVM tool. The following papers can be consulted for additional details:

- Chang and Lin, Training {nu}-Support Vector Classifiers: Theory and Algorithms
- Chih-Chung Chang and Chih-Jen Lin, LIBSVM : a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm

solve_qp3_using_smo
dlib/optimization.h
dlib/optimization/optimization_solve_qp3_using_smo_abstract.h
This function solves the following quadratic program:

       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.

This object implements the strategy used by the LIBSVM tool. The following papers can be consulted for additional details:

- Chih-Chung Chang and Chih-Jen Lin, LIBSVM : a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm
- 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.

solve_qp4_using_smo
dlib/optimization.h
dlib/optimization/optimization_solve_qp_using_smo_abstract.h
This function solves the following quadratic program:

       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.

max_cost_assignment
dlib/optimization.h
dlib/optimization/max_cost_assignment_abstract.h
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.

Note that dlib also contains a machine learning method for learning the cost function needed to use the Hungarian algorithm.

max_cost_assignment.py.html
max_sum_submatrix
dlib/optimization.h
dlib/optimization/max_sum_submatrix_abstract.h
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.
find_max_factor_graph_nmplp
dlib/optimization.h
dlib/optimization/find_max_factor_graph_nmplp_abstract.h
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:

       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.

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.
This function is an implementation of the NMPLP algorithm introduced in the following papers:

> Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations (2008) by Amir Globerson and Tommi Jaakkola

> Introduction to dual decomposition for inference (2011) by David Sontag, Amir Globerson, and Tommi Jaakkola

find_max_parse_cky
dlib/optimization.h
dlib/optimization/find_max_parse_cky_abstract.h
This function implements the CKY parsing algorithm. In particular, it finds the maximum scoring binary parse tree that parses an input sequence of tokens.
parse_tree_to_string
dlib/optimization.h
dlib/optimization/find_max_parse_cky_abstract.h
This is a set of functions useful for converting a parse tree output by find_max_parse_cky into a bracketed string suitable for displaying the parse tree.
find_trees_not_rooted_with_tag
dlib/optimization.h
dlib/optimization/find_max_parse_cky_abstract.h
Finds all the largest non-overlapping parse trees in tree that are not rooted with a particular tag.

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.

find_max_factor_graph_viterbi
dlib/optimization.h
dlib/optimization/find_max_factor_graph_viterbi_abstract.h
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.
potts_model_score
dlib/graph_cuts.h
dlib/graph_cuts/find_max_factor_graph_potts_abstract.h
This routine computes the model score for a Potts problem and a candidate labeling. This score is the quantity maximised by the find_max_factor_graph_potts routine.
graph_cut_score
dlib/graph_cuts.h
dlib/graph_cuts/min_cut_abstract.h
This routine computes the score for a candidate graph cut. This is the quantity minimized by the min_cut algorithm.
min_cut
dlib/graph_cuts.h
dlib/graph_cuts/min_cut_abstract.h
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:

> An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision, by Yuri Boykov and Vladimir Kolmogorov, in PAMI 2004.

find_max_factor_graph_potts
dlib/graph_cuts.h
dlib/graph_cuts/find_max_factor_graph_potts_abstract.h
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.

The implementation of this routine is based on the min_cut object.

oca
dlib/optimization.h
dlib/optimization/optimization_oca_abstract.h
This object is a tool for solving the following optimization problem:

       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.

For a detailed discussion you should consult the following papers from the Journal of Machine Learning Research:

> Optimized Cutting Plane Algorithm for Large-Scale Risk Minimization by Vojtech Franc, Soren Sonnenburg; 10(Oct):2157--2192, 2009.

> 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.

find_min_bobyqa
dlib/optimization.h
dlib/optimization/optimization_bobyqa_abstract.h
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.

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

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

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 find_min_single_variable function.

optimization_ex.cpp.html
model_selection_ex.cpp.html
find_max_bobyqa
dlib/optimization.h
dlib/optimization/optimization_bobyqa_abstract.h
This function is identical to the find_min_bobyqa 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.

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 find_max_single_variable function.

optimization_ex.cpp.html
model_selection_ex.cpp.html
find_min_using_approximate_derivatives
dlib/optimization.h
dlib/optimization/optimization_abstract.h
Performs an unconstrained minimization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy). This version doesn't take a gradient function but instead numerically approximates the gradient.
optimization_ex.cpp.html
solve_least_squares
dlib/optimization.h
dlib/optimization/optimization_least_squares_abstract.h
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)
least_squares_ex.cpp.html
solve_least_squares_lm
dlib/optimization.h
dlib/optimization/optimization_least_squares_abstract.h
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)
least_squares_ex.cpp.html
find_min_trust_region
dlib/optimization.h
dlib/optimization/optimization_trust_region_abstract.h
Performs an unconstrained minimization of a nonlinear function using a trust region method.
optimization_ex.cpp.html
find_max_trust_region
dlib/optimization.h
dlib/optimization/optimization_trust_region_abstract.h
Performs an unconstrained maximization of a nonlinear function using a trust region method.
find_max
dlib/optimization.h
dlib/optimization/optimization_abstract.h
Performs an unconstrained maximization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy).
find_max_box_constrained
dlib/optimization.h
dlib/optimization/optimization_abstract.h
Performs a box constrained maximization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy). This function uses a backtracking line search along with a gradient projection step to handle the box constraints.
find_max_single_variable
dlib/optimization.h
dlib/optimization/optimization_line_search_abstract.h
Performs a bound constrained maximization of a nonlinear function. The function must be of a single variable. Derivatives are not required.
find_max_using_approximate_derivatives
dlib/optimization.h
dlib/optimization/optimization_abstract.h
Performs an unconstrained maximization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy). This version doesn't take a gradient function but instead numerically approximates the gradient.