LAGRANGIAN-CONIC RELAXATIONS, PART I: A UNIFIED FRAMEWORK AND ITS APPLICATIONS TO QUADRATIC OPTIMIZATION PROBLEMS

Abstract

In Part I of a series of study on Lagrangian-conic relaxations, we introduce a unified framework for conic and Lagrangian-conic relaxations of quadratic optimization problems (QOPs) and polynomial optimization problems (POPs). The framework is constructed with a linear conic optimization problem (COP) in a finite dimensional Hilbert space, where the cone used is not necessarily convex. By imposing a copositive condition on the COP, we establish fundamental theoretical results for the COP, its (convex hull) conic relaxations, its Lagrangian-conic relaxations, and their duals. A linearly constrained QOP with complementarity constraints and a general POP can be reduced to the COP satisfying the copositivity condition. Thus the conic and Lagrangian-conic relaxations of such a QOP and POP can be discussed in a unified manner. The Lagrangian-conic relaxation takes a particularly simple form involving only a single equality constraint together with the cone constraint, which is very useful for designing efficient numerical methods. As demonstration of the elegance and power of the unified framework, we present the derivation of the completely positive programming relaxation, and a sparse doubly nonnegative relaxation for a class of a linearly constrained QOPs with complementarity constraints. The unified framework is applied to general POPs in Part II.