B-475 Lagrangian-Conic Relaxations, Part I: A Unified Framework and Its Applications to Quadratic Optimization Problems

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 con- structed with a linear conic optimization problem (COP) in a finite dimensional vector space endowed with an inner product, 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 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. Then, the conic and Lagrangian-conic relax- ations of such a QOP and POP are discussed in a unified manner. The Lagrangian-conic relaxation takes one of the simplest forms, which is very useful to design efficient numerical methods. As for applications of the framework, we discuss the completely positive pro- gramming relaxation, and a sparse doubly nonnegative relaxation for a linearly constrained QOP with complementarity constraints. The unified framework is applied to general POPs in Part II.

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