Solving Multiobjective Mixed Integer Convex Optimization Problems

Santis, Marianna De; Eichfelder, Gabriele; Niebling, Julia; Rocktäschel, Stefan

doi:10.1137/19m1264709

Cited by 30 publications

(26 citation statements)

References 31 publications

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“…In industrial applications, the number of continuous variables typically clearly dominates the number of discrete variables. Existing algorithms for bicriteria optimization of mixed-integer programs are able to compute approximations of the Pareto frontier ( [2][3][4]). Although for many of these algorithms it is known that their results converge to the true Pareto frontier, state-of-the-art methods do not provide specific bounds on the convergence speed.…”

Section: Motivation and Conceptsmentioning

confidence: 99%

“…• Algorithms based on scalarization which generates special scalarized MIPs that are then solved to optimality to obtain a new point and corresponding information about the Pareto frontier. The papers [2,18] discuss the convex case, while [3,4,19] focus on the linear case.…”

Section: Related Workmentioning

confidence: 99%

“…We use the ordering cone R K ≥0 to describe a dominance relation between two multicriteria objective vectors z (1) ∈ R K and z (2) ∈ R K . The solution z (1) is said to dominate z (2) with respect to the ordering cone R K ≥0 if z (2) ∈ z (1) + (R K ≥0 \ {0}). Let Z := f (X) ⊆ R K be the (nonempty) set of all attainable vectors in objective space.…”

Section: Bicriteria Convex Mixed-integer Problemsmentioning

confidence: 99%

“…with shared values for the discrete variables. Let z (1) ∈ f (c 1 , d) + R K ≥0 and z (2) ∈ f (c 2 , d) + R K ≥0 be two points that are contained in the upper sets of the corresponding objective vectors. Then, the line segment {λz (1) + (1 − λ)z (2) | λ ∈ [0, 1]} between z (1) and z (2) is a segment patch.…”

Section: Patchesmentioning

confidence: 99%

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An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Diessel

2021

Optimization

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Pareto frontiers of bicriteria continuous convex problems can be efficiently computed and optimal theoretical performance bounds have been established. In the case of bicriteria mixedinteger problems, the approximation of the Pareto frontier becomes, however, significantly harder. In this paper, we propose a new algorithm for approximating the Pareto frontier of bicriteria mixed-integer programs with convex constraints. Such Pareto frontiers are composed of patches of solutions with shared assignments for the discrete variables. By adaptively creating such a patchwork, our algorithm is able to create approximations that converge quickly to the true Pareto frontier. As a quality measure, we use the difference in hypervolume between the approximation and the true Pareto frontier. At least a certain number of patches is required to obtain an approximation with a given quality. This patch complexity gives a lower bound on the number of required computations. We show that our algorithm performs a number of optimization steps that are of a similar order as this lower bound. We provide an efficient MIP-based implementation of this algorithm. The efficiency of our algorithm is illustrated with numerical results showing that our algorithm has a strong theoretical performance guarantee while being competitive with other state-of-the-art approaches in practice.

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Section: Motivation and Conceptsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Bicriteria Convex Mixed-integer Problemsmentioning

confidence: 99%

Section: Patchesmentioning

confidence: 99%

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An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Diessel

2021

Optimization

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“…For specific cases (e.g. convex mixed-integer problems or non-convex continuous problems), global optimization methods have been extended to generate guarantees of global efficiency/non-dominance for multi-objective problems, using scalarization approaches (Fernández and Tóth 2007;Ehrgott and Gandibleux 2007) or multi-objective branch-and-bound (De Santis et al 2019;Niebling and Eichfelder 2019). Instead of a strict subset of the Pareto front, these methods return a set of feasible solutions and a guarantee of their non-dominance, within a given tolerance, in the form of a tight superset of the efficient and/or non-dominated sets.…”

Section: Introductionmentioning

confidence: 99%

Bi-objective design-for-control of water distribution networks with global bounds

2021

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This manuscript investigates the design-for-control (DfC) problem of minimizing pressure induced leakage and maximizing resilience in existing water distribution networks. The problem consists in simultaneously selecting locations for the installation of new valves and/or pipes, and optimizing valve control settings. This results in a challenging optimization problem belonging to the class of non-convex bi-objective mixed-integer non-linear programs (BOMINLP). In this manuscript, we propose and investigate a method to approximate the non-dominated set of the DfC problem with guarantees of global non-dominance. The BOMINLP is first scalarized using the method of $$\epsilon $$ ϵ -constraints. Feasible solutions with global optimality bounds are then computed for the resulting sequence of single-objective mixed-integer non-linear programs, using a tailored spatial branch-and-bound (sBB) method. In particular, we propose an equivalent reformulation of the non-linear resilience objective function to enable the computation of global optimality bounds. We show that our approach returns a set of potentially non-dominated solutions along with guarantees of their non-dominance in the form of a superset of the true non-dominated set of the BOMINLP. Finally, we evaluate the method on two case study networks and show that the tailored sBB method outperforms state-of-the-art global optimization solvers.

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Multi-objective branch-and-bound for the design-for-control of water distribution networks with global bounds

Ulusoy,

Stoianov

2024

Struct Multidisc Optim

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This study investigates a design-for-control (DfC) problem formulation for the joint minimization of pressure-induced leakage, maximization of resilience, and minimization of cost in water distribution networks (WDN). The DfC problem, which consists in simultaneously installing new valves and/or pipes and optimizing valve control settings, results in a challenging optimization problem belonging to the class of non-convex multi-objective mixed-integer non-linear programs (MOMINLP). Due to their complex mathematical structure, multi-objective WDN design-for-control problems have previously been solved using general-purpose evolutionary algorithms or local deterministic methods, which do not provide guarantees on the quality of the returned solutions. While branch-and-bound (BB) frameworks have been proposed to approximate the Pareto fronts of MOMINLPs with global bounds, they rely on the availability of attainable solutions which, for WDN design-for-control problems, can be hard to identify. Moreover, in the absence of general-purpose solvers, the performance of multi-objective BB implementations depends, for a given application, on the choice of adequate branching and lower bounding strategies. In this study, we investigate a multi-objective BB algorithm based on tailored branching, lower bounding and additional upper bounding strategies to efficiently approximate the Pareto front of WDN design-for-control problems with bounds of $$\epsilon$$ ϵ -non-dominance. The proposed algorithm is applied to the optimal DfC of two case study networks and is shown to outperform alternative solution methods.

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Solving Multiobjective Mixed Integer Convex Optimization Problems

Cited by 30 publications

References 31 publications

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

An adaptive patch approximation algorithm for bicriteria convex mixed-integer problems

Bi-objective design-for-control of water distribution networks with global bounds

Multi-objective branch-and-bound for the design-for-control of water distribution networks with global bounds

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