On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

Müller, Benjamin; Muñoz, Gonzalo; Gasse, Maxime; Gleixner, Ambros; Lodi, Andrea; Serrano, Felipe

doi:10.1007/978-3-030-45771-6_25

Cited by 3 publications

(3 citation statements)

References 57 publications

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“…The pricing problem becomes harder, but provides stronger dual bounds with weaker working hypothesis. The master problems must be solved with a surrogate algorithm [37] that returns optimal surrogate multipliers λ .…”

Section: Delayed Column Generationmentioning

confidence: 99%

Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes

Chicoisne

2023

Comput Optim Appl

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Solving large scale nonlinear optimization problems requires either significant computing resources or the development of specialized algorithms. For Linear Programming (LP) problems, decomposition methods can take advantage of problem structure, gradually constructing the full problem by generating variables or constraints. We first present a direct adaptation of the Column Generation (CG) methodology for nonlinear optimization problems, such that when optimizing over a structured set X plus a moderate number of complicating constraints, we solve a succession of 1) restricted master problems on a smaller set S ⊂ X and 2) pricing problems that are Lagrangean relaxations wrt the complicating constraints. The former provides feasible solutions and feeds dual information to the latter. In turn, the pricing problem identifies a variable of interest that is then taken into account into an updated subset S ⊂ X .Our approach is valid whenever the master problem has zero Lagrangean duality gap wrt to the complicating constraints, and not only when S is the convex hull of the generated variables as in CG for LPs, but also with a variety of subsets such as the conic hull, the linear span, and a special variable aggregation set. We discuss how the structure of S and its update mechanism influence the number of iterations required to reach near-optimality and the difficulty of solving the restricted master problems, and present linearized schemes that alleviate the computational burden of solving the pricing problem.We test our methods on synthetic portfolio optimization instances with up to 5 million variables including nonlinear objective functions and second order cone constraints. We show that some CGs with linearized pricing are 2-3 times faster than solving the complete problem directly and are able to provide solutions within 1% of optimality in 6 hours for the larger instances, whereas solving the complete problem runs out of memory.

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Section: Delayed Column Generationmentioning

confidence: 99%

Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes

Chicoisne

2023

Comput Optim Appl

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“…In the case of a nonempty set described by a finite number of linear inequalities we know, thanks to the Farkas Lemma, that any implied inequality can be obtained via an aggregation. Aggregations have also been studied in the context of integer programming (for example [7]) to obtain cutting-planes and in mixed-integer nonlinear programming (for example [18]) to obtain better dual bounds.…”

Section: Introductionmentioning

confidence: 99%

On obtaining the convex hull of quadratic inequalities via aggregations

Dey¹,

Muñoz²,

Serrano³

2021

Preprint

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A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran [28] showed that the convex hull of the set is given by at most two aggregated inequalities. In this work, we study the case of a set described by three or more quadratic inequalities. We show that, under technical assumptions, the convex hull of a set described by three quadratic inequalities can be obtained via (potentially infinitely many) aggregated inequalities. We also show, through counterexamples, that it is unlikely to have a similar result if either the technical conditions are relaxed, or if we consider four or more inequalities.

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“…Aggregation and disaggregation techniques provide also dual and primal bounds, used in network flow models [11] or scheduling problems [12]. We note that surrogate optimization and aggregation techniques have a recent interest in nonlinear optimization [13].…”

Section: Introductionmentioning

confidence: 99%

Machine Learning-Guided Dual Heuristics and New Lower Bounds for the Refueling and Maintenance Planning Problem of Nuclear Power Plants

Dupin

Talbi

2020

Algorithms

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This paper studies the hybridization of Mixed Integer Programming (MIP) with dual heuristics and machine learning techniques, to provide dual bounds for a large scale optimization problem from an industrial application. The case study is the EURO/ROADEF Challenge 2010, to optimize the refueling and maintenance planning of nuclear power plants. Several MIP relaxations are presented to provide dual bounds computing smaller MIPs than the original problem. It is proven how to get dual bounds with scenario decomposition in the different 2-stage programming MILP formulations, with a selection of scenario guided by machine learning techniques. Several sets of dual bounds are computable, improving significantly the former best dual bounds of the literature and justifying the quality of the best primal solution known.

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On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

Cited by 3 publications

References 57 publications

Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes

Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes

On obtaining the convex hull of quadratic inequalities via aggregations

Machine Learning-Guided Dual Heuristics and New Lower Bounds for the Refueling and Maintenance Planning Problem of Nuclear Power Plants

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