Results 1 
8 of
8
LOWER BOUNDS ON THE GLOBAL MINIMUM OF A POLYNOMIAL
"... Abstract. We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460473], to obtain a lower bound fgp,M for a multivariate polynomial f(x) ∈ R[x] of degree ≤ 2d in n variables x = (x1,..., xn) on the closed ball {x ∈ R n: ∑ x 2d i ≤ M}, computable by geometric programming, f ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460473], to obtain a lower bound fgp,M for a multivariate polynomial f(x) ∈ R[x] of degree ≤ 2d in n variables x = (x1,..., xn) on the closed ball {x ∈ R n: ∑ x 2d i ≤ M}, computable by geometric programming, for any real M. We compare this bound with the (global) lower bound fgp obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796816]. Our computations show that the bound fgp,M improves on the bound fgp and that the computation of fgp,M, like that of fgp, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down. 1.
Control and verification of highdimensional systems via DSOS and SDSOS optimization
 In Proceedings of the 53rd IEEE Conference on Decision and Control
, 2014
"... Abstract — In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to highdimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the c ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract — In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to highdimensional problems arising in control applications. Despite the wide acceptance of SOS programming in the control and optimization communities, scalability has been a key challenge due to its reliance on semidefinite programming (SDP) as its main computational engine. While SDPs have many appealing features, current SDP solvers do not approach the scalability or numerical maturity of LP and SOCP solvers. Our approach is based on the recent work of Ahmadi and Majumdar [1], which replaces the positive semidefiniteness constraint inherent in the SOS approach with stronger conditions based on diagonal dominance and scaled diagonal dominance. This leads to the DSOS and SDSOS cones of polynomials, which can be optimized over using LP and SOCP respectively. We demonstrate this approach on four high dimensional control problems that are currently well beyond the reach of SOS programming: computing a region of attraction for a 22 dimensional system, analysis of a 50 node network of oscillators, searching for degree 3 controllers and degree 8 Lyapunov functions for an Acrobot system (with the resulting controller validated on a hardware platform), and a balancing controller for a 30 state and 14 control input model of the ATLAS humanoid robot. While there is additional conservatism introduced by our approach, extensive numerical experiments on smaller instances of our problems demonstrate that this conservatism can be small compared to SOS programming. I.
A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications 1
, 2013
"... We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is b ..."
Abstract
 Add to MetaCart
(Show Context)
We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by (f ∗ −ɛ) and from above by f ∗ + ɛ(n + 1), where f ∗ is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sumsofsquares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments. 1.
A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign Structure
"... Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alter ..."
Abstract
 Add to MetaCart
(Show Context)
Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially nonpositive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sumofsquares relaxation, and so, can be solved via a single semidefinite programming problem.
LOWER BOUNDS FOR POLYNOMIALS WITH SIMPLEX NEWTON POLYTOPES BASED ON GEOMETRIC PROGRAMMING
"... ar ..."
(Show Context)
LOWER BOUNDS FOR A POLYNOMIAL ON A BASIC CLOSED SEMIALGEBRAIC SET USING GEOMETRIC PROGRAMMING
"... Abstract. Let f, g1,..., gm be elements of the polynomial ring R[x1,..., xn]. The paper deals with the general problem of computing a lower bound for f on the subset of Rn defined by the inequalities gi ≥ 0, i = 1,...,m. The paper shows that there is an algorithm for computing such a lower bound, ba ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Let f, g1,..., gm be elements of the polynomial ring R[x1,..., xn]. The paper deals with the general problem of computing a lower bound for f on the subset of Rn defined by the inequalities gi ≥ 0, i = 1,...,m. The paper shows that there is an algorithm for computing such a lower bound, based on geometric programming, which applies in a large number of cases. For example, the algorithm computes a lower bound for f on a hypercube ∏n i=1[−Ni, Ni], or, more generally, on any product of hyperellipsoids of a suitable form. The algorithm extends and generalizes earlier algorithms of Ghasemi and Marshall, dealing with the case m = 0, and of Ghasemi, Lasserre and Marshall, dealing with the case m = 1 and g1 = M − (xd1 + · · ·+ xdn). Here, d is required to be an even integer ≥ max{2, deg(f)}. The algorithm is implemented in a SAGE program developed by the first author. The bound obtained is typically not as good as the bound obtained using semidefinite programming, but it has the advantage that it is computable rapidly, even in cases where the bound obtained by semidefinite programming is not computable. 1.
1 A Perturbed Sums of Squares Theorem for Polynomial Optimization and its Applications
, 2014
"... We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is ..."
Abstract
 Add to MetaCart
(Show Context)
We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by (f¤¡) and from above by f¤+ (n+1), where f ¤ is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sumsofsquares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments. 1.