This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let ${\cal E}$ and ${\cal E}_+$ be a finite dimensional real vector space and a symmetric cone embedded in ${\cal E}$; examples of $\calE$ and $\calE_+$ include a pair of the … Read more
Results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables. Citation submitted to the special issue on Global Optimization of Math. Programming Article Download View A comparison of complete global optimization … 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
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
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
A refined version of the Karush-John first order optimality conditions is presented which reduces the number of constraints for which a constraint qualification is needed. This version is a generalization both of the Karush-John conditions and of the first order optimality conditions for concave constraints. Article Download View Sharpening the Karush-John optimality conditions
This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT specified by a set of boolean variables and a propositional formula in … Read more
We consider the problem of global minimization of rational functions on $\LR^n$ (unconstrained case), and on an open, connected, semi-algebraic subset of $\LR^n$, or the (partial) closure of such a set (constrained case). We show that in the univariate case ($n=1$), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced … Read more
We recently discovered 5 new putative globally optimum configurations for Morse clusters at $\rho=8$. This report contains some algorithmic details as well as the structures determined with our method. Citation Technical Report DSI 3-2003, Dipartimento di Sistemi e Informatica, Università degli Studi di Firenze, Firenze, 2003. Article Download View New global optima for Morse clusters … Read more