A General Framework for Convex Relaxation of Polynomial Optimization Problems over Cones
نویسندگان
چکیده
The class of POPs (Polynomial Optimization Problems) over cones covers a wide range of optimization problems such as 0-1 integer linear and quadratic programs, nonconvex quadratic programs and bilinear matrix inequalities. This paper presents a new framework for convex relaxation of POPs over cones in terms of linear optimization problems over cones. It provides a unified treatment of many existing convex relaxation methods based on the lift-and-project linear programming procedure, the reformulationlinearization technique and the semidefinite programming relaxation for a variety of problems. It also extends the theory of convex relaxation methods, and thereby brings flexibility and richness in practical use of the theory.
منابع مشابه
Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs
In this paper, under a suitable regularity condition, we establish that a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibits exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone-program, we give unified results that appl...
متن کاملA Note on Sparse SOS and SDP Realxations for Polynomial Optimization Problems over Symmetric Cones
This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal POPs. A numerical example i...
متن کاملSecond-Order Cone Relaxations for Binary Quadratic Polynomial Programs
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...
متن کاملAnalysis and Control of Partial Differential Equations using Occupation Measures
Context This work is in the line of research with the following issue: how to develop new convex optimization techniques based on semidefinite programming (SDP) and real algebraic geometry to solve optimal control problems (OCP) in a nonlinear setting. Recently, several research efforts allowed to solve numerically certain optimal control problems with polynomial data. The general idea is to re...
متن کاملAnalysis and Control of Partial Differential Equations using Occupation Measures
Context This work is in the line of research with the following issue: how to develop new convex optimization techniques based on semidefinite programming (SDP) and real algebraic geometry to solve optimal control problems (OCP) in a nonlinear setting. Recently, several research efforts allowed to solve numerically certain optimal control problems with polynomial data. The general idea is to re...
متن کامل