Global minimization of indefinite quadratic problems

Naval Research Logistics, 1993

A branch-and-bound algorithm is proposed for global minimization of indefinite quadratic functions subject to box constraints. Branching is done according to the sign of first-order derivatives. New tests based on the compatibility of signs of several first-order derivatives and on various bounding procedures, allow curtailment of the search. Computational experiments are reported. Comparison is made with an interval arithmetic implementation. 0

1988

Abstract: This paper proposes different methods for finding the global minimum of concave function subject to quadratic separable constraints. THe first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed.

International Journal of Computing Science and Mathematics, 2017

This paper addresses the optimisation of quadratic functions on convex polyhedrons. We propose a method and an algorithm for optimising a strictly concave function. This algorithm uses a good initialisation by searching the nearest vertex of a convex set to an external point in order to surround the area where the optimal solution can be located. The optimal solution may be the nearest vertex found or a boundary point obtained by the projection of the critical point onto a separating hyperplane passing through the nearest vertex. This method and this algorithm can be adapted for the convex quadratic problem. In this case, the optimal solution is the farthest vertex from the critical point.

Mathematical Programming Computation, 2018

We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In order to overcome this shortcoming, we reformulate the problem using the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LPbased branch-and-bound algorithm with extended formulations.

Operations Research, 1986

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. Subject classification: 625 globally minimizing concave functions over polytopes, 627 implicit enumeration of vertices.

Journal of Global Optimization, 1998

Researchers first examined the problem of separable concave programming more than thirty years ago, making it one of the earliest branches of nonlinear programming to be explored. This paper proposes a new algorithm that finds the exact global minimum of this problem in a finite number of iterations. In addition to proving that our algorithm terminates finitely, the paper extends a guarantee of finiteness to all branch-and-bound algorithms for concave programming that (1) partition exhaustively using rectangular subdivisions and (2) branch on the incumbent solution when possible. The algorithm uses domain reduction techniques to accelerate convergence; it solves problems with as many as 100 nonlinear variables, 400 linear variables and 50 constraints in about five minutes on an IBM RS/6000 Power PC. An industrial application with 152 nonlinear variables, 593 linear variables, and 417 constraints is also solved in about ten minutes.

Discrete Applied Mathematics, 2005

In this paper we consider the problem of optimizing a quadratic pseudo-Boolean function subject to the cardinality constraint 1 i n x i = k with a polyhedral method. More precisely we propose a study of the convex hull of feasible points included in the Padberg's Boolean quadric polytope and satisfying the cardinality constraint. Specifically, we investigate the connection with the Boolean quadric polytope and study a facet family. The relationship with two other polytopes of the literature is also explored.

Journal of Combinatorial Optimization, 1998

In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for minimizing an indefinite quadratic function over a bounded polyhedral convex set which is not necessarily given explicitly by a system of linear inequalities and/or equalities. It is required that for this set there exists an efficient algorithm to verify whether a point is feasible, and

Computational Optimization and Applications

We propose new algorithms for (i) the local optimization of bound constrained quadratic programs, (ii) the solution of general definite quadratic programs, and (iii) finding either a point satisfying given linear equations and inequalities or a certificate of infeasibility. The algorithms are implemented in Matlab and tested against state-of-the-art quadratic programming software. Keywords Definite quadratic programming • Bound constrained indefinite quadratic programming • Dual program • Certificate of infeasibility

Computational and Applied Mathematics, 2020

In this work, we propose an algorithm for finding an approximate global minimum of a concave quadratic function with a negative semi-definite matrix, subject to linear equality and inequality constraints, where the variables are bounded with finite or infinite bounds. The proposed algorithm starts with an initial extreme point, then it moves from the current extreme point to a new one with a better objective function value. The passage from one basic feasible solution to a new one is done by the construction of certain approximation sets and solving a sequence of linear programming problems. In order to compare our algorithm with the existing approaches, we have developed an implementation with MATLAB and conducted numerical experiments on numerous collections of test problems. The obtained numerical results show the accuracy and the efficiency of our approach.