A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints

Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two copies of the $m=1$ case. However, there is no known explicit, tractable, exact convex representation for $m \ge 3$. In this paper, we construct a new, polynomially sized semidefinite relaxation for all $m$, which does not employ a disjunctive approach. We show that our relaxation is exact for $m=2$. Then, for $m \ge 3$, we demonstrate empirically that it is fast and strong compared to existing relaxations. The key idea of the relaxation is a simple lifting of the original problem into dimension $n+1$. Extending this construction: (i) we show that nonconvex quadratic programming over $\|x\| \le \min \{ 1, g + h^T x \}$ has an exact semidefinite representation; and (ii) we construct a new relaxation for quadratic programming over the intersection of two ellipsoids, which globally solves all instances of a benchmark collection from the literature

Multi-objective optimization of an advanced combined cycle power plant including CO2 separation options

This paper illustrates a methodology developed to facilitate the analysis of complex systems characterized by a large number of technical, economical and environmental parameters. Thermo-economic modeling of a natural gas combined cycle including CO2 separation options has been coupled within a multi-objective evolutionary algorithm to characterize the economic and environmental performances of such complex systems within various contexts. The method has been applied to a case of power generation in Germany. The optimum options for system integration under different boundary conditions are revealed by the Pareto Optimal Frontiers. Results show the influence of the configuration and technical parameters on the electrical efficiencies of the Pareto optimal plants and their sub-systems. The results provide information on the relationship between power generation cost and CO2 emissions, and allow sensitivity analyses of important economical parameters like natural gas and electricity prices. Such a tool is of interest for power generation technology suppliers, for utility owners or for project investors, and for policy makers in the context of CO2 mitigation schemes including emission trading

Continuous relaxation of MINLP problems by penalty functions: a practical comparison

A practical comparison of penalty functions for globally solving mixed-integer nonlinear programming (MINLP) problems is presented. The penalty approach relies on the continuous relaxation of the MINLP problem by adding a specific penalty term to the objective function. A new penalty algorithm that addresses simultaneously the reduction of the error tolerances for optimality and feasibility, as well as the reduction of the penalty parameter, is designed. Several penalty terms are tested and different penalty parameter update schemes are analyzed. The continuous nonlinear optimization problem is solved by the deterministic DIRECT optimizer. The numerical experiments show that the quality of the produced solutions are satisfactory and that the selected penalties have different performances in terms of efficiency and robustness.This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundação para a Ciência e Tecnologia, within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio