The main contributions in this thesis are advances in parametric programming. The thesis is divided into two parts; theoretical advances and application areas for parametric programming. The first part deals with continuity properties and the structure of solutions to convex parametric quadratic and linear programs. The second part focuses on applications of parametric quadratic and linear prog...

The problem of obtaining the maximum a posteriori estimate of a general discrete random field (i.e. a random field defined using a finite and discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximate algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on conve...

The paper describes a class of mathematical problems at an intersection of operator theory and combinatorics, and discusses their application in complex system analysis. The main object of study is duality gap bounds in quadratic programming which deals with problems of maximizing quadratic functionals subject to quadratic constraints. Such optimization is known to be universal, in the sense th...

Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, secondorder cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our framework, we re-derive previous relaxation schemes and provide new ones. In particular, w...

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic prog...

In this paper, we give optimality conditions for the quadratic programming problems with constraints defined by finitely many convex in Hilbert spaces. As special cases, obtain under linear

In this paper, we analyze in depth a simplicial decomposition like algorithmic framework for large scale convex quadratic programming. In particular, we first propose two tailored strategies for handling the master problem. Then, we describe a few techniques for speeding up the solution of the pricing problem. We report extensive numerical experiments on both real portfolio optimization and gen...

In this paper, the problem under consideration is multiobjective non-linear fractional programming problem involving semilocally convex and related functions. We have discussed the interrelation between the solution sets involving properly efficient solutions of multiobjective fractional programming and corresponding scalar fractional programming problem. Necessary and sufficient optimality...

Abstract In this paper, the notions of subgradmnt, subdifferentla[, and differential with respect to convex fuzzy mappings are investigated, whmh provides the basis for the fuzzy extremum problem theory We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optlmahty conditions. Furthermore, the concept of saddle...

Newton’s method for solving variational inequalities is known to be locally quadratically convergent. By incorporating a line search strategy for the regularized gap function, Taji et al. (Mathematical Programming, 1993) have proposed a modification of a Newton’s method which is globally convergent and whose rate of convergence is quadratic. But the quadratic convergence has been shown only und...