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 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 non-positive 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 sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW preprint, 22 page

Limit analysis and inf-sup conditions on convex cones

This paper is focused on analysis and reliable computations of limit loads in perfect plasticity. We recapitulate our recent results arising from a continuous setting of the so-called limit analysis problem. This problem is interpreted as a convex optimization subject to conic constraints. A related inf-sup condition on a convex cone is introduced and its importance for theoretical and numerical purposes is explained. Further, we introduce a penalization method for solving the kinematic limit analysis problem. The penalized problem may be solved by standard finite elements due to available convergence analysis using a simple local mesh adaptivity. This solution concept improves the simplest incremental method of limit analysis based on a load parametrization of an elastic-perfectly plastic problem

Limit analysis and inf-sup conditions on convex cones

This paper is focused on analysis and reliable computations of limit loads in perfect plasticity. We recapitulate our recent results arising from a continuous setting of the so-called limit analysis problem. This problem is interpreted as a convex optimization subject to conic constraints. A related inf-sup condition on a convex cone is introduced and its importance for theoretical and numerical purposes is explained. Further, we introduce a penalization method for solving the kinematic limit analysis problem. The penalized problem may be solved by standard finite elements due to available convergence analysis using a simple local mesh adaptivity. This solution concept improves the simplest incremental method of limit analysis based on a load parametrization of an elastic-perfectly plastic problem

Gordon's inequality and condition numbers in conic optimization

The probabilistic analysis of condition numbers has traditionally been approached from different angles; one is based on Smale's program in complexity theory and features integral geometry, while the other is motivated by geometric functional analysis and makes use of the theory of Gaussian processes. In this note we explore connections between the two approaches in the context of the biconic homogeneous feasiblity problem and the condition numbers motivated by conic optimization theory. Key tools in the analysis are Slepian's and Gordon's comparision inequalities for Gaussian processes, interpreted as monotonicity properties of moment functionals, and their interplay with ideas from conic integral geometry

A D-induced duality and its applications

This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone D, and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the D-induced duality in the paper. Basic properties of the extended duality, including the extended bi-polar theorem, are proven. Examples are give to show the applications of the new results.Duality;Convex cones;Bi-polar theorem;Conic optimization

A D-induced duality and its applications

This paper attempts to extend the notion of duality for convex cones, by basing it on a predescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone D, and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the D-induced duality in the paper. We further introduce the notion of D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced polar sets within the same framework, which can be viewed as a generalization of the D-induced dual cones and are convenient to use for some practical applications. Properties of the extended duality, including the extended bi-polar theorem, are proven. Furthermore, attention is paid to the computation and approximation of the D-induced dual objects. We discuss, as examples, applications of the newly introduced D-induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization.bi-polar theorem;conic optimization;convex cones;duality

Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.Comment: NIPS 201

Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability

We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation