Convexifying of polynomials by convex factor

W pracy podajemy nowe wyniki dotyczące "uwypuklania" wielomianów rzeczywistych, a w szczególności uogólniamy niektóre z rezultatów z pracy K. Kurdyka, S. Spodzieja, „Convexifying positive polynomials and sums of squares approximation”, SIAM J. Optim. 25 (2015), no. 4, 2512-2536. Pokazujemy też pewne zastosowania uzyskanych wyników w optymalizacji

On the Effective Putinar's Positivstellensatz and Moment Approximation

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraicset S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Lojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition),this Lojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre's Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo-moment sequences, which are positive on the quadratic module of S

용량 제약이 없는 부보상 문제의 혼합 이진 이차 문제로의 모형화를 통한 해법

학위논문(석사) -- 서울대학교대학원 : 공과대학 산업공학과, 2022. 8. 홍성필.부보상 문제는 비순환 유향 그래프 상에서 출발, 도착 마디를 잇는 경로와 그 경로 상의 흐름을 결정하는 문제이다. 부보상은 도시 1에서 n까지 이동하면서 각 도시를 경유하 거나 지나치며, 경유하는 도시에서만 상품을 매매할 수 있지만 이동 거리와 상품량에 따른 비용 또한 지불해야 한다. 이 때, 부보상은 자신의 수익, 즉 총 상품의 판매량에서 얻는 수익과 지불 비용의 차를 최대화하고자 한다. 본 연구에서는 용량 제약이 없는 경우만을 다루며 기존 부보상 문제를 혼합이진이차문제으로 재모형화하여 분지절단법 으로 문제를 푼다. 이 때 목적 함수를 볼록화하고 연속 완화시켜 얻을 수 있는 상한을 비교하기 위해 여러 볼록화 방법들을 비교하고 비교실험한 결과 또한 제시한다.Bubosang Problem is a problem set on a directed acyclic graph path concerning both the path and multi-commodity flow decisions. A merchant travels from city 1 through n, either transiting through a city and trading products or passing by the city to the next city on his route. He wants to choose the path and trading product quantity to maximize his net profit which is defined by the difference between the total sales revenue and the traveling cost. The scope of the study considers only the uncapacitated case. In this study, we reformulate BP into a mixed binary quadratic problem to employ the branch-and-cut algorithm to solve the problem. Specifically, we compare the upper bound obtained through the continuous relaxation and convexification of the objective by studying different convexification methods. Computational results of the comparison are also provided.Chapter 1 Introduction 1 1.1 Background 1 1.2 Literature Review 3 1.3 Research Motivations 5 1.4 Organization of the Thesis 6 Chapter 2 Problem Definition and Mathematical Formulations 7 2.1 Problem Definition 7 2.2 Flow Arc Formulation 8 2.3 MBQP Formulation 11 2.3.1 MBQP 13 2.4 Branch-and-Cut Algorithm 14 2.4.1 Overall Setting 14 2.4.2 Cutset Inequality 14 2.4.3 Lower Bound 15 2.4.4 Upper Bound 18 Chapter 3 Convexification Methods 19 3.1 One Coefficient Case : Eigenvalue Method 21 3.2 Criteria for Convexification Evaluation 22 3.2.1 Criterion for Unweighted Methods 22 3.3 Two Coefficient Case : (α, β) - SDP method 23 3.4 Two Coefficient Case : (α, β) - Sum of Squares Method 24 3.5 Four Coefficient Case : (α, β, γ, δ) - method 26 3.5.1 (α, β, γ, δ) - SDP method 26 3.5.2 (α, β, γ, δ) - Sum of Squares method 28 3.6 Five Coefficient Case : (α, β, γ, δ, τ ) - Sum of Squares method 29 3.7 Weighted methods 30 3.7.1 Criterion for Weighted Methods 30 Chapter 4 Computational Experiments 32 Chapter 5 Conclusion 36 Bibliography 37 국문초록 41석

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

Error bounds for monomial convexification in polynomial optimization

Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of $[0,1]^n$. This implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over $[-1,1]^n$.Comment: 33 pages, 2 figures, to appear in journa