Multi-Criteria Optimization

Multi-Objective Optimization for Politically Fair Districting: A Scalable Multilevel Approach

  • Rahul Swamy
  • Douglas King
  • Sheldon Jacobson
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences, Multi-Criteria Optimization Tags , , , , , , ,

    Political districting in the United States is a decennial process of redrawing the boundaries of congressional and state legislative districts. The notion of fairness in political districting has been an important topic of subjective debate, with district maps having consequences to multiple stakeholders. Even though districting as an optimization problem has been well-studied, existing models … Read more

    Subdifferentials and SNC property of scalarization functionals with uniform level sets and applications

  • Christiane Tammer
  • Bao Q. Truong
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization, Nonsmooth Optimization

    This paper deals with necessary conditions for minimal solutions of constrained and unconstrained optimization problems with respect to general domination sets by using a well-known nonlinear scalarization functional with uniform level sets (called Gerstewitz’ functional in the literature). The primary objective of this work is to establish revised formulas for basic and singular subdifferentials of … Read more

    Learning to Project in Multi-Objective Binary Linear Programming

  • Hadi Charkhgard
  • Iman Dayarian
  • Ali Eshragh
  • Sorna Javadi
  • Alvaro Sierra-Altamiranda
    • Categories (Mixed) Integer Linear Programming, Multi-Criteria Optimization, Other Topics Tags , , , ,

    In this paper, we investigate the possibility of improving the performance of multi-objective optimization solution approaches using machine learning techniques. Specifically, we focus on multi-objective binary linear programs and employ one of the most effective and recently developed criterion space search algorithms, the so-called KSA, during our study. This algorithm computes all nondominated points of … Read more

    The Sard theorem for essentially smooth locally Lipschitz maps and applications in optimization

  • Xuan Duc Ha Truong
    • Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization Tags , , , ,

    The classical Sard theorem states that the set of critical values of a $C^{k}$-map from an open set of $\R^n$ to $\R^p$ ($n\geq p$) has Lebesgue measure zero provided $k\geq n-p+1$. In the recent paper by Barbet, Dambrine, Daniilidis and Rifford, the so called “preparatory Sard theorem” for a compact countable set $I$ of $C^k$ … Read more

    On prime and minimal representations of a face of a polyhedron

  • Ta Van Van Tu
    • Categories Combinatorial Optimization, Multi-Criteria Optimization, Polyhedra

    In this paper, a new method for determining all minimal representations of a face of a polyhedron is proposed. A main difficulty for determining prime and minimal representations of a face is that the deletion of one redundant constraint can change the redundancy of other constraints. To reduce computational efforts in finding all minimal representations … Read more

    An Algorithmic Approach to Multiobjective Optimization with Decision Uncertainty

  • Gabriele Eichfelder
  • Julia Niebling
  • Stefan Rocktäschel
    • Categories Global Optimization, Multi-Criteria Optimization, Robust Optimization Tags , ,

    In real life applications optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one … Read more

    On the extension of the Hager-Zhang conjugate gradient method for vector optimization

  • Max L. N. Gonçalves
  • Leandro Fonseca Prudente
    • Categories Multi-Criteria Optimization

    The extension of the Hager-Zhang (HZ) nonlinear conjugate gradient method for vector optimization is discussed in the present research. In the scalar minimization case, this method generates descent directions whenever, for example, the line search satisfies the standard Wolfe conditions. We first show that, in general, the direct extension of the HZ method for vector … Read more

    Performance indicators in multiobjective optimization

  • Charles Audet
  • Jean Bigeon
  • Dominique Cartier
  • Sébastien Le Digabel
  • Ludovic Salomon
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization

    In recent years, the development of new algorithms for multiobjective optimization has considerably grown. A large number of performance indicators has been introduced to measure the quality of Pareto front approximations produced by these algorithms. In this work, we propose a review of a total of 63 performance indicators partitioned into four groups according to … Read more

    Approximations for Pareto and Proper Pareto solutions and their KKT conditions

  • Kalyanmoy Deb
  • Joydeep Dutta
  • Poonam Kesarwani
  • Pradyumn Kumar Shukla
    • Categories Multi-Criteria Optimization Tags , , ,

    There has been numerous amount of studies on proper Pareto points in multiobjective optimization theory. Geoffrion proper points are one of the most prevalent form of proper optimality. Due to some convergence issues a restricted version of these proper points, Geoffrion proper points with preset bounds has been introduced recently. Since solution of any algorithm … Read more

    Nonmonotone line searches for unconstrained multiobjective optimization problems

  • Ellen H. Fukuda
  • Kanako Mita
  • Nobuo Yamashita
    • Categories Multi-Criteria Optimization, Unconstrained Optimization Tags , , , ,

    In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the … Read more