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In this paper, we propose a new convergent conic programming hierarchy ofrelaxations involving both semi-definite cone and second-order cone constraintsfor solving nonconvex polynomial optimization problems to global optimality.The significance of this hierarchy is that the size and number of thesemi-definite and second-order cone constraints of the relaxations are fixedand independent of the step or level of the approximation in the hierarchy.Using the Krivine-Stengle's certificate of positivity in real algebraicgeometry, we establish the convergence of the hierarchy of relaxations,extending the very recent so-called bounded degree Lasserre hierarchy. Inparticular, we also provide a convergent bounded degree second-order coneprogramming (SOCP) hierarchy for solving polynomial optimization problems. Wethen present finite convergence at step one of the SOCP hierarchy for twoclasses of polynomial optimization problems: a subclass of convex polynomialoptimization problems where the objective and constraint functions areSOCP-convex polynomials, defined in terms of specially structured sum ofsquares polynomials, and a class of polynomial optimization problems, involvingpolynomials with essentially non-positive coefficients. In the case of one-stepconvergence for problems with SOCP-convex polynomials, we show how a globalsolution is recovered from the relaxation via Jensen's inequality ofSOCP-convex polynomials. As an application, we derive a correspondingconvergent conic linear programming hierarchy for conic-convex semi-algebraicprograms. Whenever the semi-algebraic set of the conic-convex program isdescribed by convex polynomial inequalities, we show further that the values ofthe relaxation problems converge to the common value of the convex program andits Lagrangian dual under a constraint qualification.

A Bounded Degree Lasserre Hierarchy with SOCP Relaxations for Global Polynomial Optimization and Applications