PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

Hartikainen, Markus; Lovison, Alberto

doi:10.1007/s10898-014-0232-9

Cited by 14 publications

(17 citation statements)

References 35 publications

(50 reference statements)

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“…The sectorization of water distribution systems has also provided some benefit in this respect. Networks are subdivided into smaller sectors, called District Metered Areas (DMAs), so that flow into and out of each DMA is continuously monitored, improving the management of pressure and leakage (Farley and Trow 2003). On the other hand, this has severely reduced network redundancy, affecting resilience and water quality negatively.…”

Section: Introductionmentioning

confidence: 99%

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

Pecci

Abraham

Stoianov

2016

Struct Multidisc Optim

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In this paper, we study the multiobjective codesign problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model.

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Section: Introductionmentioning

confidence: 99%

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

Pecci

Abraham

Stoianov

2016

Struct Multidisc Optim

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“…In first-order techniques, piecewise linear functions are used to approximate the POF [9], in second-order techniques, piecewise quadratic functions are used to approximate the POF [10], and so on. An alternative classification is based on the applicability of the approximation methods, such as convex [11][12][13], nonconvex [14], or both [15][16][17][18][19]. There also exist several inexact surrogate models for approximation [20], such as radial basis functions [21], Kriging [22], and artificial neural networks [23].…”

Section: Introductionmentioning

confidence: 99%

“…Notable work in this direction is the PAINT algorithm presented in Refs. [2], [16], and [18]. Its applicability was also demonstrated on two computationally expensive problems: heat exchanger (HE) network synthesis [15] and a wastewater treatment problem [17].…”

Section: Introductionmentioning

confidence: 99%

A Projection-Based Approach for Constructing Piecewise Linear Pareto Front Approximations

Singh

Bhattacharjee

Ray

2016

Journal of Mechanical Design

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Real-life design problems often require simultaneous optimization of multiple conflicting criteria resulting in a set of best trade-off solutions. This best trade-off set of solutions is referred to as Pareto optimal front (POF) in the outcome space. Obtaining the complete POF becomes impractical for problems where evaluation of each solution is computationally expensive. Such problems are commonly encountered in several fields, such as engineering, management, and scheduling. A practical approach in such cases is to construct suitable POF approximations, which can aid visualization, decision-making, and interactive optimization. In this paper, we propose a method to generate piecewise linear Pareto front approximation from a given set of N Pareto optimal outcomes. The approximations are represented using geometrical linear objects known as polytopes, which are formed by triangulating the given M-objective outcomes in a reduced ðM À 1Þ-objective space. The proposed approach is hence referred to as projection-based Pareto interpolation (PROP). The performance of PROP is demonstrated on a number of benchmark problems and practical applications with linear and nonlinear fronts to illustrate its strengths and limitations. While being novel and theoretically interesting, PROP also improves on the computational complexity required in generating such approximations when compared with existing Pareto interpolation (PAINT) algorithm.

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“…These components can superimpose and intersect each other when they are mapped on the Pareto front as in the example represented in Figure 1(b). Only in very special situations, i.e., pathological cases, the set of Pareto optima is a unique vector, or a finite number of vectors [10,12,11,13]. The explicit definition and the details of this function, called L&H 2x2 , are given in [11].…”

mentioning

confidence: 99%

“…Only in very special situations, i.e., pathological cases, the set of Pareto optima is a unique vector, or a finite number of vectors [10,12,11,13]. The explicit definition and the details of this function, called L&H 2x2 , are given in [11]. The Pareto set is obtained by the sicon method [12].…”

mentioning

confidence: 99%

Exact extension of the DIRECT algorithm to multiple objectives

Lovison¹,

Miettinen²

2019

AIP Conference Proceedings

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The direct algorithm has been recognized as an efficient global optimization method which has few requirements of regularity and has proven to be globally convergent in general cases. direct has been an inspiration or has been used as a component for many multiobjective optimization algorithms. We propose an exact and as genuine as possible extension of the direct method for multiple objectives, providing a proof of global convergence (i.e., a guarantee that in an infinite time the algorithm becomes everywhere dense). We test the efficiency of the algorithm on a nonlinear and nonconvex vector function.

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PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

Cited by 14 publications

References 35 publications

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

Scalable Pareto set generation for multiobjective co-design problems in water distribution networks: a continuous relaxation approach

A Projection-Based Approach for Constructing Piecewise Linear Pareto Front Approximations

Exact extension of the DIRECT algorithm to multiple objectives

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