Further Results on Approximating Nonconvex Quadratic Optimizationby Semidefinite Programming Relaxation
We study approximation bounds for the SDP relaxation of quadratically constrained quadratic optimization: min f^0(x) subject to f^k(x)
A sequential convexification method (SCM) for continuous global optimization
A new method for continuous global minimization problems, acronymed SCM, is introduced. This method gives a simple transformation to convert the original objective function to an auxiliary function with gradually fewer local minimizers. All Local minimizers except a prefixed one of the auxiliary function is in the region where the function value of the original … Read more
A new class of proximal algorithms for the nonlinear complementarity problem
In this paper, we consider a variable proximal regularization method for solving the nonlinear complementarity problem for P0 functions. CitationApplied Optimization Series, 96, Optimization and Control With Applications, L. Qi, K. Teo and X. Yang (Eds.), pp 549-561, Springer, 2005.
A Dynamic Large-Update Primal-Dual Interior-Point Method for Linear Optimization
Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy … Read more
Semidefinite programming vs LP relaxations for polynomial programming
We consider the global minimization of a multivariate polynomial on a semi-algebraic set \Omega defined with polynomial inequalities. We then compare two hierarchies of relaxations, namely, LP-relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams and recent SDP (semi definite programming) relaxations introduced by the … Read more
An explicit equivalent positive semidefinite program for nonlinear 0-1 programs
We consider the general nonlinear optimization problem in 0-1 variables and provide an explicit equivalent positive semidefinite program in $2^n-1$ variables. The optimal values of both problems are identical. From every optimal solution of the former one easily find an optimal solution of the latter and conversely, from every solution of the latter one may … Read more
A Laplace transform algorithm for the volume of a convex polytope
We provide two algorithms for computing the volume of the convex polytope $\Omega:=\{x\in \R^n_+ \,|\,Ax\leq b\}$, for $A\in\R^{m\times n}, b\in\R^n$. The computational complexity of both algorithms is essentially described by $n^m$, which makes them especially attractive for large $n$ and relatively small $m$, when the other methods with $O(m^n)$ complexity fail. The methodology which differs … Read more
Power transmission network design by a greedy randomized adaptive path relinking approach
This paper illustrates results obtained by a new metaheuristic approach, Greedy Randomized Adaptive Path Relinking, applied to solve static power transmission network design problems. This new approach consists of a generalization of GRASP concepts to explore different trajectories between two CitationAT&T Labs Research Report, December 2001 Submitted to PSCC’02.ArticleDownload View PDF
A Comparative Study of Large-Scale Nonlinear Optimization Algorithms
In recent years, much work has been done on implementing a variety of algorithms in nonlinear programming software. In this paper, we analyze the performance of several state-of-the-art optimization codes on large-scale nonlinear optimization problems. Extensive numerical results are presented on different classes of problems, and features of each code that make it efficient or … Read more
Smoothing Method of Multipliers for Sum-Max Problems
We study nonsmooth unconstrained optimization problem, which includes sum of pairwise maxima of smooth functions. Minimum $l_1$-norm approximation is a particular case of this problem. Combining ideas Lagrange multipliers with smooth approximation of max-type function, we obtain a new kind of nonquadratic augmented Lagrangian. Our approach does not require artificial variables, and preserves sparse structure … Read more