James E. Falk

Jointly Constrained Biconvex Programming

This paper presents a branch-and-bound algorithm for minimizing the sum of a convex function in x, a convex function in y and a bilinear term in x and y over a closed set. Such an objective function is called biconvex with biconcave functions similarly defined. The feasible region of this model permits joint constraints in x and y to be expressed. The bilinear programming problem becomes a special case of the problem addressed in this paper. We prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point. The algorithm is proven to converge to a global solution of the nonconvex program. We discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs, for which the algorithm has been implemented.

An explicit solution to the multi-level programming problem☆

Abstract The multi-level programming problem is defined as an n-person nonzero-sum game with perfect information in which the players move sequentially. The bi-level linear case is addressed in detail. Solutions are obtained by recasting this problem as a standard mathematical probram and appealing to its implicitly separable structure. The reformulated optimization problem is linear save for a complementarity constraint of the form 〈u, g〉 = 0. This constraint is decomposed in a manner that permits us to achieve separability with very little cost in dimensionality. A general branch and bound algorithm is then applied to obtain solutions. Unlike the conventional mathematical program though, the multi-level program may fail to have a solution even when the decision variables are defined over a compact set. An auxiliary optimization problem is employed to detect such failure. Finally, the general max-min problem is discussed within the bi-level programming framework. Examples are given for a variety of related problems.

A Successive Underestimation Method for Concave Minimization Problems

A new method designed to globally minimize concave functions over linear polyhedra is described. Properties of the method are discussed, an example problem is solved, and computational considerations are discussed.

A linear max—min problem

We consider a two person max—min problem in which the maximizing player moves first and the minimizing player has perfect information of the outcome of this move. The move of the maximizing player influences not only the objective function but also the constraints of the minimizing player. The joint constraints as well as the objective function are assumed to be linear.For this problem it is shown that the familiar inequality min max ⩾ max min is reversed due to the influence of the joint constraints. The problem is characterized as a nonconvex program and a method of solution based on the branch and bound philosophy is given. A small example is presented to illlustrate the algorithm.

Infinitely Constrained Optimization Problems.

A generalized cutting-plane algorithm designed to solve problems of the form min{f(x) :x ∈X andg(x,y) ∈ 0 for ally ∈Y} is described. Convergence is established in the general case (f,g continuous,X andY compact). Constraint dropping is allowed in a special case [f,g(·,y) convex functions,X a convex set]. Applications are made to a variety of max-min problems. Computational considerations are discussed.

On bilevel programming, part I: general nonlinear cases

This paper is concerned with general nonlinear nonconvex bilevel programming problems (BLPP). We derive necessary and sufficient conditions at a local solution and investigate the stability and sensitivity analysis at a local solution in the BLPP. We then explore an approach in which a bundle method is used in the upper-level problem with subgradient information from the lower-level problem. Two algorithms are proposed to solve the general nonlinear BLPP and are shown to converge to regular points of the BLPP under appropriate conditions. The theoretical analysis conducted in this paper seems to indicate that a sensitivity-based approach is rather promising for solving general nonlinear BLPP.

Image space analysis of generalized fractional programs

The solution of a particular nonconvex program is usually very dependent on the structure of the problem. In this paper we identify classes of nonconvex problems involving either sums or products of ratios of linear terms which may be treated by analysis in a transformed space. In each class, the image space is defined by a mapping which associates a new variable with each original ratio of linear terms. In the image space, optimization is easy in certain directions, and the overall solution may be realized by sequentially optimizing in these directions.In addition to these ratio problems, we also show how to use image space analysis to treat the subclass of problems whose objective is to optimize a product of linear terms. For each class of nonconvex problems, we present an algorithm that locates global solutions by computing both upper and lower bounds on the solution and then solving a sequence of linear programming sub-problems. We also demonstrate the algorithms described in this paper by solving several example problems.

Concave minimization via collapsing polytopes

We present a procedure for globally minimizing a concave function over a bounded polytope by successively minimizing the function over polytopes containing the feasible region, and collapsing to the feasible region. The initial containing polytope is a simplex, and, at the kth iteration, the procedure chooses the most promising vertex of the current containing polytope to refine the approximation. The method generates a tree whose ultimate terminal nodes coincide with the vertices of the feasible region, and accounts for the vertices of the containing polytopes.

Integer Prim-Read Solutions to a Class of Target Defense Problems

We study the choice of a deployment and firing doctrine for defending separated point targets of potentially different values against an attack by an unknown number of sequentially arriving missiles. We minimize the total number of defenders, subject to an upper bound on the maximum expected value damage per attacking weapon. We show that the Greedy Algorithm produces an optimal integral solution to this problem.

A Strategic Weapons Exchange Allocation Model

We formulate a two-strike strategic weapons exchange, model as a max-min problem with the first strikers allocation affecting the second strikers feasible region. The max-min problem is shown to be equivalent to a separable, nonconvex program, to which an algorithm designed to locate an approximate global solution is then applied. The solution of three example problems is given.