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.

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# backtracking_line_search

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.
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#include <dlib/optimization.h>
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# bfgs_search_strategy

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.

C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# call_function_and_expand_args

This routine allows you to pass a `dlib::matrix<double,0,1>` object to a function that takes simple `double` arguments. It does this by automatically expanding the matrix elements and invoking the function. For example, suppose you had a function like this:

    double f(double x, double y, double z);

You could then call f() like this:

    matrix<double,0,1> args = {3,4,5};
    call_function_and_expand_args(f, args); // calls: f(3,4,5)

This kind of thing is convenient when writing optimizers like find_max_global 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.
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#include <dlib/global_optimization.h>
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# cg_search_strategy

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).

C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# clamp_function

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.
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#include <dlib/optimization.h>
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# derivative

This is a function that takes another function as input and returns a function object that numerically computes the derivative of the input function.
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#include <dlib/optimization.h>
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# elastic_net

This object is a tool for solving the following optimization problem:

       min_w:      length_squared(X*w - Y) + ridge_lambda*length_squared(w)
       such that:  sum(abs(w)) <= lasso_budget

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.

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.

The particular algorithm used to solve it is described in the paper:

> 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

And for the SVM solver sub-component we use the algorithm from:

> 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.

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#include <dlib/optimization/elastic_net.h>
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# find_gap_between_convex_hulls

This function measures the position and size of the gap between two convex polytopes. In particular, it solves the following quadratic program:

       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

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#include <dlib/optimization.h>
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# find_max

Performs an unconstrained maximization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy).
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#include <dlib/optimization.h>
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# find_max_bobyqa

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.

C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# find_max_box_constrained

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.
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#include <dlib/optimization.h>
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# find_max_factor_graph_nmplp

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

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#include <dlib/optimization.h>
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# find_max_factor_graph_potts

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.

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#include <dlib/graph_cuts.h>
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# find_max_factor_graph_viterbi

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.
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#include <dlib/optimization.h>
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# find_max_global

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 global_function_search 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.
C++ Example Programs: optimization_ex.cpp, model_selection_ex.cpp
Python Example Programs: global_optimization.py
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#include <dlib/global_optimization.h>
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# find_max_parse_cky

This function implements the CKY parsing algorithm. In particular, it finds the maximum scoring binary parse tree that parses an input sequence of tokens.
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#include <dlib/optimization.h>
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# find_max_single_variable

Performs a bound constrained maximization of a nonlinear function. The function must be of a single variable. Derivatives are not required.
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#include <dlib/optimization.h>
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# find_max_trust_region

Performs an unconstrained maximization of a nonlinear function using a trust region method.
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#include <dlib/optimization.h>
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# find_max_using_approximate_derivatives

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.
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#include <dlib/optimization.h>
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# find_min

Performs an unconstrained minimization of a nonlinear function using some search strategy (e.g. bfgs_search_strategy).
C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# find_min_bobyqa

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.

C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# find_min_box_constrained

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.
C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# find_min_global

This function is identical to the find_max_global 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.
C++ Example Programs: optimization_ex.cpp, model_selection_ex.cpp
Python Example Programs: global_optimization.py
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#include <dlib/global_optimization.h>
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# find_min_single_variable

Performs a bound constrained minimization of a nonlinear function. The function must be of a single variable. Derivatives are not required.
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#include <dlib/optimization.h>
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# find_min_trust_region

Performs an unconstrained minimization of a nonlinear function using a trust region method.
C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# find_min_using_approximate_derivatives

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.
C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# find_trees_not_rooted_with_tag

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.

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#include <dlib/optimization.h>
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# global_function_search

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.

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:

1. A global model that upper bounds our objective function. 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 Global Optimization of Lipschitz Functions by Cédric Malherbe and Nicolas Vayatis in the 2017 International Conference on Machine Learning.
2. A local quadratic model fit around the best point seen so far. This uses a trust region method similar to what is proposed in: The NEWUOA software for unconstrained optimization without derivatives By M.J.D. Powell, 40th Workshop on Large Scale Nonlinear Optimization (Erice, Italy, 2004)

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.

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.

Video of optimizer running

Finally, note that the find_max_global 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. So if find_max_global() isn't general enough for you then you can use the global_function_search API.

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#include <dlib/global_optimization.h>
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# gradient_norm_stop_strategy

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.
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#include <dlib/optimization.h>
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# graph_cut_score

This routine computes the score for a candidate graph cut. This is the quantity minimized by the min_cut algorithm.
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#include <dlib/graph_cuts.h>
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# isotonic_regression

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.

This is done using the fast O(n) pool adjacent violators algorithm.

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#include <dlib/optimization.h>
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# lagrange_poly_min_extrap

This function finds the second order polynomial that interpolates a set of points and returns the minimum of that polynomial.
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#include <dlib/optimization.h>
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# lbfgs_search_strategy

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.

C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# line_search

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.
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#include <dlib/optimization.h>
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# make_line_search_function

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.
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#include <dlib/optimization.h>
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# max_cost_assignment

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.

C++ Example Programs: max_cost_assignment_ex.cpp
Python Example Programs: max_cost_assignment.py
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#include <dlib/optimization.h>
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# max_sum_submatrix

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.
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#include <dlib/optimization.h>
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# min_cut

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.

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#include <dlib/graph_cuts.h>
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# mpc

This object implements a linear model predictive controller. In particular, it solves a certain quadratic program using the method described in the paper:

> A Fast Gradient method for embedded linear predictive control (2011) by Markus Kogel and Rolf Findeisen

C++ Example Programs: mpc_ex.cpp
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#include <dlib/control.h>
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# negate_function

This is a function that takes another function as input and returns a function object that computes the negation of the input function.
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#include <dlib/optimization.h>
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# newton_search_strategy

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.

C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# objective_delta_stop_strategy

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.
C++ Example Programs: optimization_ex.cpp
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#include <dlib/optimization.h>
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# oca

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.

    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.

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.

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#include <dlib/optimization.h>
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# parse_tree_to_string

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.
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#include <dlib/optimization.h>
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# poly_min_extrap

This function finds the 2nd or 3rd degree polynomial that interpolates a set of points and returns the minimum of that polynomial.
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#include <dlib/optimization.h>
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# potts_model_score

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.
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#include <dlib/graph_cuts.h>
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# solve_least_squares

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)
C++ Example Programs: least_squares_ex.cpp
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#include <dlib/optimization.h>
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# solve_least_squares_lm

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)
C++ Example Programs: least_squares_ex.cpp
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#include <dlib/optimization.h>
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# solve_qp2_using_smo

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

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#include <dlib/optimization.h>
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# solve_qp3_using_smo

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.

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#include <dlib/optimization.h>
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# solve_qp4_using_smo

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 - 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.

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#include <dlib/optimization.h>
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# solve_qp_box_constrained

This function solves the following quadratic program:

       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.

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#include <dlib/optimization.h>
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# solve_qp_box_constrained_blockdiag

This function solves the following quadratic program:

       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.

So it does the same thing as solve_qp_box_constrained, 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.
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#include <dlib/optimization.h>
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# solve_qp_using_smo

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.

More Details...
#include <dlib/optimization.h>
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# solve_trust_region_subproblem

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

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#include <dlib/optimization.h>
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# solve_trust_region_subproblem_bounded

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
       lower(i) <= p(i) <= upper(i), for all i

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#include <dlib/optimization.h>
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# upper_bound_function

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.

This is based on the method described in Global Optimization of Lipschitz Functions 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.

More Details...
#include <dlib/global_optimization.h>