Feasibility-Based Bounds Tightening via Fixed Points

Belotti, Pietro; Cafieri, Sonia; Lee, Jon; Liberti, Leo

doi:10.1007/978-3-642-17458-2_7

Cited by 32 publications

(47 citation statements)

References 30 publications

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“…Clearly as |I i | and |I j | approach ∞, the error terms of the McCormick envelopes, E( l i , l j ) and E( l i , u j ), approach zero, and thus enforcing x i = x * i , 4 Exhaustiveness of the partitioning scheme implies AMP will eventually partition all other domains small enough such that AMP will pick an active partition with the global optimal whose length is ≤ l i + u i .…”

Section: Computing Lower and Upper Boundsmentioning

confidence: 99%

An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

Nagarajan

Wang

et al. 2019

J Glob Optim

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In this work, we develop an adaptive, multivariate partitioning algorithm for solving nonconvex, Mixed-Integer Nonlinear Programs (MINLPs) with polynomial functions to global optimality. In particular, we present an iterative algorithm that exploits piecewise, convex relaxation approaches via disjunctive formulations to solve MINLPs that is different than conventional spatial branch-and-bound approaches. The algorithm partitions the domains of variables in an adaptive and non-uniform manner at every iteration to focus on productive areas of the search space. Furthermore, domain reduction techniques based on sequential, optimization-based bound-tightening and piecewise relaxation techniques, as a part of a presolve step, are integrated into the main algorithm. Finally, we demonstrate the effectiveness of the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare our algorithm with state-of-the-art global optimization solvers. With our novel approach, we solve several largescale instances, some of which are not solvable by state-of-the-art solvers. We also succeed in reducing the best known optimality gap for a hard, generalized pooling problem instance.

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Section: Computing Lower and Upper Boundsmentioning

confidence: 99%

An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

Nagarajan

Wang

et al. 2019

J Glob Optim

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“…A recent method by Belotti [71] uses pairs of constraints instead of individual constraints to infer bounds. Different techniques have been developed to infer bounds on MILP problems [62,63], and on MINLP problems [64,72]. To illustrate this method consider the constraint x 3 = T and x up = (4, 2, 4) T .…”

Section: Feasibility-based Bound Tighteningmentioning

confidence: 99%

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

Trespalacios

Grossmann

2014

Chemie Ingenieur Technik

152

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This work presents a review of the applications of mixed-integer nonlinear programming (MINLP) in process systems engineering (PSE). A review on the main deterministic MINLP solution methods is presented, including an overview of the main MINLP solvers. Generalized disjunctive programming (GDP) is an alternative higher-level representation of MINLP problems. This work reviews some methods for solving GDP models, and techniques for improving MINLP methods through GDP. The paper also provides a high-level review of the applications of MINLP in PSE, particularly in process synthesis, planning and scheduling, process control and molecular computing.

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“…This, however, is rarely true if bound tightening techniques [26,18,19,27] are used as a preprocessing step; and it is never true during sBB with rectangular partitioning schemes, as the variable ranges are partitioned at each node. We also emphasize that the remark given in [8] p. 11 is also valid in our setting: for polynomial degrees ≥ 3, the proposed convex relaxation might not be monotonically increasing w.r.t.…”

Section: Choosing a Good Basis For The Companion Systemmentioning

confidence: 99%

Compact Relaxations for Polynomial Programming Problems

Cafieri

Hansen

Létocart³

et al. 2012

Experimental Algorithms

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Abstract. Reduced RLT constraints are a special class of Reformulation-Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-integer) quadratic programming problems subject to systems of linear equality constraints. We present an extension to the general case of polynomial programming problems and discuss the derived convex relaxation. We then show how to perform rRLT constraint generation so as to reduce the number of inequality constraints in the relaxation, thereby making it more compact and faster to solve. We present some computational results validating our approach.

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Feasibility-Based Bounds Tightening via Fixed Points

Cited by 32 publications

References 30 publications

An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

Compact Relaxations for Polynomial Programming Problems

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