The so-called S-lemma has played an important role in optimization, both in theory and in applications. The significance of S-lemma is especially pronounced in control theory, robust optimization, and non-convex quadratic optimization. Hitherto, S-lemma is however established only in the domain of quadratic functions. In this paper we shall extend the notion of S-lemma to the class of univariat...

The quadratic assignment problem (denoted QAP), in the trace formulation over the permutation matrices, is min X2 tr(AXB + C)X t : Several recent lower bounds for QAP are discussed. These bounds are obtained by applying continuous optimization techniques to approximations of this combinatorial optimization problem, as well as by exploiting the special matrix structure of the problem. In particu...

Stochastic differential equations (SDEs) have been applied by engineers and economists because it can express the behavior of stochastic processes in compact expressions. In this paper, by using Grunwald-Letnikov fractional derivative, the stochastic differential model is improved. Two numerical examples are presented to show efficiency of the proposed model. A numerical optimization approach b...

Quadratic assignment problem is one of the great challenges in combinatorial optimization. It has many applications in Operations research and Computer Science. In this paper, the author extends the most-used rounding approach to a one-parametric optimization model for the quadratic assignment problems. A near-optimum parameter is also predestinated. The numerical experiments confirm the effici...

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadr...

QP is the optimization of a quadratic function subject to linear equality and inequality constraints. It arises in multiple objective decision making where the departure of the actual decisions from their corresponding ideal, or bliss, value can be evaluated using a weighted quadratic norm as a measure of deviation. The formulation of mean-variance optimization of uncertain systems also leads t...

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indelinite quadratic constraints. Under a suitable simultaneous diagonalization assumption {which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem ...

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of objective quadratic functions and diagonal coeffici...

We consider the class of max-min and min-max optimization problems subject to a global budget constraint. We undertake a systematic algorithmic and complexity-theoretic study of such problems, which we call problems design problems. Every optimization problem leads to a natural design problem. Our main result uses techniques of Freund-Schapire [FS99] from learning theory, and its generalization...