Generalized polynomial programming

Blau, Gary; Wilde, Douglass J.

doi:10.1002/cjce.5450470401

Cited by 38 publications

(10 citation statements)

References 4 publications

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“…The most prominent use of signomials in applied mathematics is signomial programming -the minimization of a signomial subject to finitely many signomial inequality constraints. Nonconvex signomial programs have been used in chemical engineering since the 1970's [3,20,23]. More recently, there has been a surge of interest in nonconvex signomial programming for aerospace engineering and transportation systems; see [29,54,72] for academic work on this topic and [37] for an industrial example.…”

Section: Why Study Signomials?mentioning

confidence: 99%

“…Its design includes careful consideration to the concept of "A-degree" (which we study in Section 3) and offers improved efficiency and stronger bounds relative to earlier SAGE-based methods. We demonstrate our approach with three worked examples: a toy nonconvex quadratic program in five variables, a problem adapted from chemical reaction network theory [48,55], and a chemical reactor design problem [3,4]. The first two of these problems are polynomial in t = exp x and so are amenable to the sums of squares (SOS) based Lasserre hierarchy [39]; see also [56].…”

Section: Contributionsmentioning

confidence: 99%

“…Consider the univariate case A = [1/4; 1/2; 1/3; 0] ∈ R 4×1 . The signomial f (x) = exp(x) admits several polynomializations, among them p 1 (y) = y 2 1 y 2 and p 2 (y) = y 3 3 . Note that p 1 ∼ = p 2 on the variety {y ∈ R 4 : y 2 1 = y 2 , y 2 2 = y 3 3 }.…”

Section: 2mentioning

confidence: 99%

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Algebraic perspectives on signomial optimization

Dressler¹,

Murray²

2021

Preprint

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Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of "degree" that is central to polynomial optimization theory. We reclaim that principle here through the concept of signomial rings, which we use to derive complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. The Positivstellensatz underlying the lower bounds relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity and Archimedeanity of the feasible set. Through worked examples we illustrate the practicality of this hierarchy in areas such as chemical reaction network theory and chemical engineering. These examples include comparisons to direct global solvers (e.g., BARON and ANTIGONE) and the Lasserre hierarchy (where appropriate). The completeness of our hierarchy of upper bounds follows from a generic construction whereby a Positivstellensatz for signomial nonnegativity over a compact set provides for arbitrarily strong outer approximations of the corresponding cone of nonnegative signomials. While working toward that result, we prove basic facts on the existence and uniqueness of solutions to signomial moment problems.

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Section: Why Study Signomials?mentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

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Algebraic perspectives on signomial optimization

Dressler¹,

Murray²

2021

Preprint

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“…A signomial optimization geometric programming technique often provides a much more accurate and better mathematical representation of real-world non-linear optimization problems. Passy and Wilde (1967) and Blau and Wilde (1969) generalized some of the prototype concepts and theorems in order to treat signomial geometric programming problems. In 1988 general type of signomial geometric programming has been done by Charnes and Cooper, who proposed methods for approximating signomial geometric programs with prototype geometric programs.…”

Section: Introductionmentioning

confidence: 99%

Modified signomial geometric programming (MSGP) and its applications

Mandal

Islam

2019

Ind. Jour. Manag. & Prod.

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A "signomial" is a mathematical function, contains one or more independent variables. Richard J. Duffin and Elmor L. Peterson introduced the term "signomial". Signomial geometric programming (SGP) optimization technique often provides a much better mathematical result of real-world nonlinear optimization problems. In this research paper, we have proposed unconstrained and constrained signomial geometric programming (SGP) problem with positive or negative integral degree of difficulty. Here a modified form of signomial geometric programming (MSGP) has been developed and some theorems have been derived. Finally, these are illustrated by proper examples and applications.

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“…MISO was originally proposed when convex posynomial geometric programs could not model important engineering applications [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Misener

Floudas

2013

J Optim Theory Appl

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Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO[1], this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulationlinearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers.

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Generalized polynomial programming

Abstract: Conventional design procedures use physical reasoning to construct equations describing the system. To find the the values o B the primal and quasidual objcctive functions are

Cited by 38 publications

References 4 publications

Algebraic perspectives on signomial optimization

Algebraic perspectives on signomial optimization

Modified signomial geometric programming (MSGP) and its applications

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

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