Reduced RLT constraints are a special class of Reformulation-Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-integer) quadratic programming problems subject to systems of linear equality constraints. We present an extension to the general case of polynomial programming problems and discuss the derived convex relaxation. We then show how to perform r...

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...

We prove that integer programming with three alternating quantifiers is NPcomplete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes P,Q ⊂ R, c...

The business environment is full of uncertainty. Allocating the wealth among various asset classes may lower the risk of overall portfolio and increase the potential for more benefit over the long term. In this paper, we propose a mixed singlestage R&D projects and multi-stage securities portfolio selection model. Specifically, we present a bi-objective mixed-integer stochastic programming mode...

Optimization problems involving two decision makers at two different decision levels are referred to as bilevel programming problems. In this work, we present a novel algorithm for the exact and global solution of two classes of bi-level programming problems, namely (i) bi-level mixed-integer linear programming problems (B-MILP) and (ii) bi-level mixed-integer quadratic programming problems (B-...

Recent adaptive memory and evolutionary metaheuristics for mixed integer programming have included proposals for introducing inequalities and target objectives to guide the search. These guidance approaches are useful in intensification and diversification strategies related to fixing subsets of variables at particular values, and in strategies that use linear programming to generate trial solu...

Bayesian games offer a suitable framework for where the utility degrees are additive in essence. This approach does nevertheless not apply to ordinal games, do capture more than ranking, nor situations of decision under qualitative uncertainty. The present paper proposes representation possibilistic incomplete information and extends fundamental notions pure mixed Nash equilibrium this framewor...