Relative Entropy Relaxations for Signomial Optimization

Signomial programs (SPs) are optimization problems specified in terms of signomials, which are weighted sums of exponentials composed with linear functionals of a decision variable. SPs are non-convex optimization problems in general, and families of NP-hard problems can be reduced to SPs. In this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of SPs. This sequence of lower bounds is computed by solving increasingly larger-sized relative entropy optimization problems, which are convex programs specified in terms of linear and relative entropy functions. Our approach relies crucially on the observation that the relative entropy function -- by virtue of its joint convexity with respect to both arguments -- provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic-geometric-mean inequality. By appealing to representation theorems from real algebraic geometry, we show that our sequences of lower bounds converge to the global optima for broad classes of SPs. Finally, we also demonstrate the effectiveness of our methods via numerical experiments

Relative Entropy Relaxations for Signomial Optimization

Signomial programs (SPs) are optimization problems specified in terms of signomials, which are weighted sums of exponentials composed with linear functionals of a decision variable. SPs are nonconvex optimization problems in general, and families of NP-hard problems can be reduced to SPs. In this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of SPs. This sequence of lower bounds is computed by solving increasingly larger-sized relative entropy optimization problems, which are convex programs specified in terms of linear and relative entropy functions. Our approach relies crucially on the observation that the relative entropy function, by virtue of its joint convexity with respect to both arguments, provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic-geometric-mean inequality. By appealing to representation theorems from real algebraic geometry, we show that our sequences of lower bounds converge to the global optima for broad classes of SPs. Finally, we also demonstrate the effectiveness of our methods via numerical experiments

Relative Entropy Relaxations for Signomial Optimization

Signomial programs (SPs) are optimization problems specified in terms of signomials, which are weighted sums of exponentials composed with linear functionals of a decision variable. SPs are nonconvex optimization problems in general, and families of NP-hard problems can be reduced to SPs. In this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of SPs. This sequence of lower bounds is computed by solving increasingly larger-sized relative entropy optimization problems, which are convex programs specified in terms of linear and relative entropy functions. Our approach relies crucially on the observation that the relative entropy function, by virtue of its joint convexity with respect to both arguments, provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic-geometric-mean inequality. By appealing to representation theorems from real algebraic geometry, we show that our sequences of lower bounds converge to the global optima for broad classes of SPs. Finally, we also demonstrate the effectiveness of our methods via numerical experiments

Signomial and polynomial optimization via relative entropy and partial dualization

We describe a generalization of the Sums-of-AM/GM-Exponential (SAGE) methodology for relative entropy relaxations of constrained signomial and polynomial optimization problems. Our approach leverages the fact that SAGE certificates conveniently and transparently blend with convex duality, in a way which enables partial dualization of certain structured constraints. This more general approach retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), and inspires a projective method of solution recovery which respects partial dualization. We illustrate the utility of our methodology with a range of examples from the global optimization literature, along with a publicly available software package

Signomial and Polynomial Optimization via Relative Entropy and Partial Dualization

We describe a generalization of the Sums-of-AM/GM Exponential (SAGE) relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that relative entropy based SAGE certificates conveniently and transparently blend with convex duality, in a manner that Sums-of-Squares certificates do not. This more general approach not only retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), but also inspires a novel perspective-based method of solution recovery. We illustrate the utility of our methodology with a range of examples from the global optimization literature, along with a publicly available software package.Comment: Software at https://rileyjmurray.github.io/sageopt/. Nine tables, one figure. Forty pages (with large margins). Ten pages of computational experiments; print pages 1-25 and 36-40 to skip the computational experiments. Version 2: minor simplification to section 4.2.

Sublinear Circuits and the Constrained Signomial Nonnegativity Problem

Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets $X$. We introduce the $X$-circuits of a finite subset $\mathcal{A} \subset \mathbb{R}^n$, which generalize the simplicial circuits of the affine-linear matroid induced by $\mathcal{A}$ to a constrained setting. The $X$-circuits exhibit particularly rich combinatorial properties for polyhedral $X$, in which case the set of $X$-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of $X$-circuits transparently reveals when an $X$-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler $X$-nonnegative signomials. We develop a duality theory for $X$-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when $X$ is polyhedral. In conjunction with a notion of reduced $X$-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization.Comment: 30 pages. V2: The title is new, and Sections 1 and 2 have been rewritten. Section 1 contains a summary of our results. We improved one result, and consolidated some other