Abstract
The bounded degree sum-of-squares (BSOS) hierarchy of Lasserre et al. (EURO J Comput Optim 1–31, 2015) constructs lower bounds for a general polynomial optimization problem with compact feasible set, by solving a sequence of semi-definite programming (SDP) problems. Lasserre, Toh, and Yang prove that these lower bounds converge to the optimal value of the original problem, under some assumptions. In this paper, we analyze the BSOS hierarchy and study its numerical performance on a specific class of bilinear programming problems, called pooling problems, that arise in the refinery and chemical process industries.
Highlights
We analyze the bounded degree sum-of-squares (BSOS) hierarchy and study its numerical performance on a specific class of bilinear programming problems, called pooling problems, that arise in the refinery and chemical process industries
The simplest way of dealing with equality constraints, is to replace each equality constraint by two inequalities; this process increases the number of constraints which is not favorable for the BSOS hierarchy
We describe how one can reduce the number of linear variables and constraints at each level of the BSOS hierarchy significantly
Summary
In 2015, Lasserre et al introduced the so-called bounded degree sum-of-squares (BSOS) hierarchy to obtain a nondecreasing sequence of lower bounds on the optimal value of problem (1) when the feasible set is compact. In a seminal paper in 2000, Lasserre (2001) first introduced a hierarchy of lower bounds for polynomial optimization using SDP relaxations. The second way of obtaining an SDP problem is called the ‘sampling formulation’, and was first studied in Lofberg and Parrilo (2004) It was used for the numerical BSOS hierarchy calculations in Lasserre et al (2015), with a set of s(τ ) randomly generated points in Rn. Δ(n, τ ) = x ∈ Rn τ x ∈ Nn, xi ≤ 1. By letting yi j be the flow from node i to node j, ui j the restriction on yi j that can be carried from i to j , and plk the concentration value of kth specification in the pool l, the pooling problem can be written as the following optimization model: min y,p ci j yi j (i, j)∈A (8)
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