Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.

DC programming: overview

We investigate the convex–concave procedure, a local heuristic that utilizes the tools of convex optimization to find local optima of difference of convex (DC) programming problems. The class of DC problems includes many difficult problems such as the traveling salesman problem. We extend the standard procedure in two major ways and describe several variations. First, we allow for the algorithm to be initialized without a feasible point. Second, we generalize the algorithm to include vector inequalities. We then present several examples to demonstrate these algorithms.

/pdf/variations-and-extension-of-the-convex-concave-procedure-4eeqw5u9xa.pdf

Variations and extension of the convex–concave procedure

In this bibliography main directions of research as well as main fields of applications of bilevel programming problems and mathematical programs with equilibrium constraints are summarized. Focus is also on the difficulties arising from nonuniqueness of lower-level optimal solutions and on optimality conditions.

Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints

This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.

A Branch-and-Reduce Approach to Global Optimization

An algorithm is described for solving large-scale nonlinear programs whose objective and constraint functions are smooth and continuously differentiable The algorithm is of the projected Lagrangian type, involving a sequence of sparse, linearly constrained subproblems whose objective functions include a modified Lagrangian term and a modified quadratic penalty function

A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints

Reverse convex programs generally have disconnected feasible regions. Basic solutions are defined and properties of the latter and of the convex hull of the feasible region are derived. Solution procedures are discussed and a cutting plane algorithm is developed.

Reverse convex programming

A constraintg(x)⩾0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.

Linear programs with an additional reverse convex constraint

A method of constructing test problems for linear bilevel programming problems is presented. The method selects a vertex of the feasible region, ‘far away’ from the solution of the relaxed linear programming problem, as the global solution of the bilevel problem. A predetermined number of constraints are systematically selected to be assigned to the lower problem. The proposed method requires only local vertex search and solutions to linear programs.

Test problem construction for linear bilevel programming problems

A modification of Tuy's cone splitting algorithm for minimizing a concave function subject to linear inequality constraints is shown to be convergent by demonstrating that the limit of a sequence of constructed convex polytopes contains the feasible region. No geometric tolerance parameters are required.

Convergence of a Tuy-type algorithm for concave minimization subject to linear inequality constraints

A new algorithm is presented for minimizing a linear function subject to a set of linear inequalities and one additional reverse convex constraint. The algorithm utilizes a conical partition of the convex polytope in conjuction with its facets in order to remain on the level surface of the reverse convex constraint. The algorithm does not need to solve linear programs on a set of cones which converges to a line segment.

Stephen E. Jacobsen

Papers

Reverse convex programming

Linear programs with an additional reverse convex constraint

Test problem construction for linear bilevel programming problems

Convergence of a Tuy-type algorithm for concave minimization subject to linear inequality constraints

A level set algorithm for a class for reverse convex programs