This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost coefficients, determine a cost-coefficient vector such that the corresponding optimal objective value of the linear program is closest to the given value. The above problem, referred here as the inverse optimal value … Read more
We theorically study, on a distributed memory architecture, the parallelization of Hansen’s algorithm for the continuous global optimization with inequality constraints, using interval arithmetic. We propose a parallel algorithm based on a dynamic redistribution of the working list among the processors. On the other hand, we exploit the reduction technique, developped by Hansen, for computing … Read more
We describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex, not necessarily differentiable function of several variables. The method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and it is related to the use of the concept … Read more
An exact-penalty-function-based scheme—inspired from an old idea due to Mayne and Polak (Math. Prog., vol.~11, 1976, pp.~67–80)—is proposed for extending to general smooth constrained optimization problems any given feasible interior-point method for inequality constrained problems. It is shown that the primal-dual interior-point framework allows for a simpler penalty parameter update rule than that discussed and … Read more
Recently Gilbert, Gonzaga and Karas [2001] constructed examples of ill-behaved central paths for convex programs. In this paper we show that under mild conditions the central path has sufficient smoothness to allow construction of a path-following interior point algorithm for non-differentiable convex programs. We show that starting from a point near the center of the … Read more
This paper presents smoothing heuristics for an NP-hard combinatorial problem based on Lagrangian relaxation. We formulate the Lagrangian dual for this nonconvex quadratic problem and propose eigenvalue nonsmooth unconstrained optimization to solve the dual problem with bundle or subgradient methods. Derived heuristics are considered to obtain good primal solutions through pathfollowing methods using a projected … Read more
We discuss in this paper a class of nonsmooth functions which can be represented, in a neighborhood of a considered point, as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector. We argue that this is a sufficiently rich class of functions and … Read more
Given vectors $a,c\in Z^n$ and $b\in Z$, we consider the (unbounded) knapsack optimization problem $P:\,\min\{c’x\,\vert\, a’x=b;\,x\in N^n\}$. We compute the minimum value $p^*$ using techniques from complex analysis, namely Cauchy residue technique to integrate a function in $C^2$, the $Z$-transform of an appropriate function related to $P$. The computational complexity depends on $s:=\sum_{a_j} a_j$, not … Read more
Hierarchies of semidefinite relaxations for $0/1$ polytopes have been constructed by Lasserre (2001a) and by Lov\’asz and Schrijver (1991), permitting to find the cut polytope of a graph on $n$ nodes in $n$ steps. We show that $\left\lceil {n\over 2} \right\rceil$ iterations are needed for finding the cut polytope of the complete graph $K_n$. CitationMathematics … Read more
Since the Motzkin-Straus result on the clique number of graphs, published in 1965, where they show that the size of the largest clique in a graph can be obtained by solving a quadratic programming problem, several results on the continuous approach to the determination of the clique number of a graph or, equivalently, to the … Read more