[top]# 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.

[top]# 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 [top]# 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 [top]# 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.

[top]# 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.

[top]# 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.

[top]# 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

[top]# find_max

Performs an unconstrained maximization of a nonlinear function using
some search strategy (e.g.

bfgs_search_strategy).

[top]# 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,

model_selection_ex.cpp [top]# 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.

[top]# 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

[top]# 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.

[top]# 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.

[top]# 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.

[top]# 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.

[top]# find_max_trust_region

Performs an unconstrained maximization of a nonlinear function using
a trust region method.

[top]# 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.

[top]# 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,

model_selection_ex.cpp [top]# 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 [top]# 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.

[top]# find_min_trust_region

Performs an unconstrained minimization of a nonlinear function using
a trust region method.

C++ Example Programs:

optimization_ex.cpp [top]# 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 [top]# find_optimal_parameters

Performs a constrained minimization of a function and doesn't require derivatives from the user.
This function is similar to

find_min_bobyqa and

find_min_single_variable except that it
allows any number of variables and never throws exceptions when the max iteration
limit is reached (even if it didn't converge).

[top]# 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.

[top]# 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.

[top]# graph_cut_score

This routine computes the score for a candidate graph cut. This is the
quantity minimized by the

min_cut algorithm.

[top]# lagrange_poly_min_extrap

This function finds the second order polynomial that interpolates a
set of points and returns the minimum of that polynomial.

[top]# 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 [top]# 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.

[top]# 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.

[top]# 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.cppPython Example Programs:

max_cost_assignment.py [top]# 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.

[top]# 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.

[top]# 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 [top]# 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.

[top]# 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 [top]# 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 [top]# 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.

[top]# 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.

[top]# 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.

[top]# 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.

[top]# 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 [top]# 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 [top]# 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

[top]# 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.

[top]# 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.

[top]# 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.

[top]# 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.

[top]# 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.

[top]# 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