We consider here the problem of minimizing a polynomial function on $\oR^n$. The problem is known to be hard even for degree $4$. Therefore approximation algorithms are of interest. Lasserre \cite{lasserre:2001} and Parrilo \cite{Pa02a} have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We … Read more
Sufficient optimality criteria for linearly constrained, concave minimization problems is given in this paper. Our optimality criteria is based on the sensitivity analysis of the relaxed linear programming problem. Our main result is similar to that of Phillips and Rosen (1993), however our proofs are simpler and constructive. Phillips and Rosen (1993) in their paper … Read more
This paper presents a new method for solving global optimization problems. We use a local technique based on the notion of discrete gradients for finding a cone of descent directions and then we use a global cutting angle algorithm for finding global minimum within the intersection of the cone and the feasible region. We present … Read more
In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined … Read more
We consider the integer program P -> max {c’x | Ax=y; x\in N^n }. Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables … Read more
This paper presents a fuzzy control approach for improving convergence time in stochastic global minimization algorithms .We show concrete results when the method is applied to an efficient algorithm based on ideas related to Simulated Annealing. ArticleDownload View PDF
A polynomial SDP (semidefinite program) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmetric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the … Read more
It is widely believed that in order to solve large scale global optimization problems an appropriate mixture of local approximation and global exploration is necessary. Local approximation, if first order information on the objective function is available, is efficiently performed by means of local optimization methods. Unfortunately, global exploration, in absence of some kind of … Read more
This survey covers the state of the art of techniques for solving general purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably find all solutions (if there are finitely many). The core of the material is presented in sufficient detail that the survey may serve as a … Read more
Sequences of generalized Lagrangian duals and their SOS (sums of squares of polynomials) relaxations for a POP (polynomial optimization problem) are introduced. Sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the … Read more