A Convergent Hierarchy of Certificates for Constrained Signomial Positivity

Optimization is at the heart of many engineering problems. Many optimization problems, however, are computationally intractable. One approach to tackle such intractability is to find a tractable problem whose solution, when found, approximates that of the original problem. Specifically, convex optimization problems are often efficiently solvable, and finding a convex formulation that approximates a nonconvex problem, known as convex relaxation, is an effective approach. This work concerns a particular class of optimization problem, namely constrained signomial optimization. Based on the idea that optimization of a function is equivalent to verifying its positivity, we first study a certificate of signomial positivity over a constrained set, which finds a decomposition of the signomial into sum of parts that are verifiably positive via convex constraints. However, the certificate only provides a sufficient condition for positivity. The main contribution of the work is to show that by multiplying additionally more complex functions, larger subset of signomials that are positive over a compact convex set, and eventually all, may be certified by the above method. The result is analogous to classic Positivstellensatz results from algebraic geometry which certifies polynomial positivity by finding its representation with sum of square polynomials. The result provides a convergent hierarchy of certificate for signomial positivity over a constrained set that is increasingly more complete. The hierarchy of certificate in turn gives a convex relaxation algorithm that computes the lower bounds of constrained signomial optimization problems that are increasingly tighter at the cost of additional computational complexity. At some finite level of the hierarchy, we obtain the optimal solution

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

Applications of Convex Analysis to Signomial and Polynomial Nonnegativity Problems

Here is a question that is easy to state, but often hard to answer: Is this function nonnegative on this set? When faced with such a question, one often makes appeals to known inequalities. One crafts arguments that are sufficient to establish the nonnegativity of the function, rather than determining the function's precise range of values. This thesis studies sufficient conditions for nonnegativity of signomials and polynomials. Conceptually, signomials may be viewed as generalized polynomials that feature arbitrary real exponents, but with variables restricted to the positive orthant. Our methods leverage efficient algorithms for a type of convex optimization known as relative entropy programming (REP). By virtue of this integration with REP, our methods can help answer questions like the following: Is there some function, in this particular space of functions, that is nonnegative on this set? The ability to answer such questions is extremely useful in applied mathematics. Alternative approaches in this same vein (e.g., methods for polynomials based on semidefinite programming) have been used successfully as convex relaxation frameworks for nonconvex optimization, as mechanisms for analyzing dynamical systems, and even as tools for solving nonlinear partial differential equations. This thesis builds from the sums of arithmetic-geometric exponentials or SAGE approach to signomial nonnegativity. The term "exponential" appears in the SAGE acronym because SAGE parameterizes signomials in terms of exponential functions. Our first round of contributions concern the original SAGE approach. We employ basic techniques in convex analysis and convex geometry to derive structural results for spaces of SAGE signomials and exactness results for SAGE-based REP relaxations of nonconvex signomial optimization problems. We frame our analysis primarily in terms of the coefficients of a signomial's basis expansion rather than in terms of signomials themselves. The effect of this framing is that our results for signomials readily transfer to polynomials. In particular, we are led to define a new concept of SAGE polynomials. For sparse polynomials, this method offers an exponential efficiency improvement relative to certificates of nonnegativity obtained through semidefinite programming. We go on to create the conditional SAGE methodology for exploiting convex substructure in constrained signomial nonnegativity problems. The basic insight here is that since the standard relative entropy representation of SAGE signomials is obtained by a suitable application of convex duality, we are free to add additional convex constraints into the duality argument. In the course of explaining this idea we provide some illustrative examples in signomial optimization and analysis of chemical dynamics. The majority of this thesis is dedicated to exploring fundamental questions surrounding conditional SAGE signomials. We approach these questions through analysis frameworks of sublinear circuits and signomial rings. These sublinear circuits generalize simplicial circuits of affine-linear matroids, and lead to rich modes of analysis for sets that are simultaneously convex in the usual sense and convex under a logarithmic transformation. The concept of signomial rings lets us develop a powerful signomial Positivstellensatz and an elementary signomial moment theory. The Positivstellensatz provides for an effective hierarchy of REP relaxations for approaching the value of a nonconvex signomial minimization problem from below, as well as a first-of-its-kind hierarchy for approaching the same value from above. In parallel with our mathematical work, we have developed the sageopt python package. Sageopt drives all the examples and experiments used throughout this thesis, and has been used by engineers to solve high-degree polynomial optimization problems at scales unattainable by alternative methods. We conclude this thesis with an explanation of how our theoretical results affected sageopt's design.</p

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

Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability

Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies