In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which r...

Various mathematical programming methods are developed in the literature for solving quadratic objective optimization problem. Conventional approaches such as Lambda iteration method, gradient method and quadratic programming method are used to solve the quadratic objective optimization problems. These conventional methods have different drawbacks of their own. In Lambda iteration method, it is...

• First treatment of standard quadratic optimization problems under uncertainty with two-stage stochastic approach. Efficient and tractable upper lower bounds for this NP-hard problem. Modern first-order method (pairwise Frank-Wolfe) enhanced copositive techniques aim at scalability bounds. Two-stage decision are characterized by the property that in stage one part has to be made before relevan...

We consider a fractional program with both linear and quadratic equation in numerator and denominator  having second order cone (SOC) constraints. With a suitable change of variable, we transform the problem into a  second order cone programming (SOCP)  problem.  For the quadratic fractional case, using a relaxation, the problem is reduced to a semi-definite optimization (SDO) program. The p...

Multidimensional scaling is a technique for exploratory analysis of multidimensional data. The essential part of the technique is minimization of a function with unfavorable properties like multimodality, invariants and non-differentiability. Recently various two-level optimization algorithms for multidimensional scaling with city-block distances have been proposed exploiting piecewise quadrati...

This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive-style problem, which itself can be...

Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices, or its conic dua...

We consider optimization problems on Riemannian manifolds with equality and inequality constraints, which we call nonlinear (RNLO) problems. Although they have numerous applications, the existing studies them are limited especially in terms of algorithms. In this paper, propose sequential quadratic (RSQO) that uses a line-search technique an $\ell_{1}$ penalty function as extension standard SQO...