Linearly constrained global optimization: a general solution algorithm with applications

• Published in: Applied mathematics and computation Vol. 134; no. 2; pp. 345 - 361
• Main Authors: Arsham, Hossein, Gradisar, Miro, Indihar Stemberger, Mojca
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 2003; Elsevier
• Subjects: Calculus of variations and optimal control; Exact sciences and technology; Global optimization; Linearly constrained optimization; Mathematical analysis; Mathematics; Nonlinear programming; Partial differential equations; Polyhedra; Sciences and techniques of general use; Linearly constrained optimization; Global optimization; Polyhedra; Nonlinear programming; Lagrangian; Maximization; Unconstrained optimization; Constraint; Parametric representation; Optimization method; Numerical method; Critical point; Nonlinear problems; Non linear programming; Algorithm; Global solution; Partial differential equation; Optimization; Constrained optimization; Polyhedron; Linearity; Non linear model; Feasibility; Objective function
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(01)00289-2

This paper presents an efficient enumerative approach to solve general linearly constrained optimization problems. This class of optimization problems includes fractional, nonlinear network models, quadratic, convex and non-convex programs. The unified approach is accomplished by converting the constrained optimization problem to an unconstrained optimization problem through a parametric representation of its feasible region. The proposed solution algorithm consists of three phases. In phase 1 it finds all interior critical points. In phase 2 the parametric representation of the feasible region is constructed to identify any critical points on the edges and faces of the feasible region. This is done by a modified version of an algorithm for finding the V-representation of the polyhedron. Then, in phase 3, the global optimal value of the objective function is found by evaluating the objective function at the critical points as well as at the vertices. For an illustration of the algorithm and a comparison with the existing methods small-size numerical examples are presented.