Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers

T. D. Chuong () and V. Jeyakumar ()
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T. D. Chuong: University of New South Wales
V. Jeyakumar: University of New South Wales

Journal of Global Optimization, 2018, vol. 72, issue 4, No 3, 655-678

Abstract: Abstract In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.

Keywords: Nonconvex polynomial programs; Generalized Lagrangian duality; Global optimality; Sum of squares polynomials; Quadratic programs; Separable programs; 49K99; 65K10; 90C29; 90C46 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10898-018-0665-7

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