multiobjective optimization

Computing an enclosure for multiobjective mixed-integer nonconvex optimization problems using piecewise linear relaxations

  • Moritz Link
  • Stefan Volkwein
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization, Network Optimization Tags , , , ,

    In this paper, a new method for computing an enclosure of the nondominated set of multiobjective mixed-integer problems without any convexity requirements is presented. In fact, our criterion space method makes use of piecewise linear relaxations in order to bypass the nonconvexity of the original problem. The method chooses adaptively which level of relaxation is … Read more

    Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems

  • Everton Silva
  • Elizabeth Wegner Karas
  • Lucelina Santos
    • Categories Multi-Criteria Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness … Read more

    Handling of constraints in multiobjective blackbox optimization

  • Jean Bigeon
  • Sébastien Le Digabel
  • Ludovic Salomon
    • Categories Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    This work proposes the integration of two new constraint-handling approaches into the blackbox constrained multiobjective optimization algorithm DMulti-MADS, an extension of the Mesh Adaptive Direct Search (MADS) algorithm for single-objective constrained optimization. The constraints are aggregated into a single constraint violation function which is used either in a two-phase approach, where research of a feasible … Read more

    A solver for multiobjective mixed-integer convex and nonconvex optimization

  • Gabriele Eichfelder
  • Oliver Stein
  • Leo Warnow
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Multi-Criteria Optimization Tags , , , , ,

    This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are … Read more

    A quasi-Newton method with Wolfe line searches for multiobjective optimization

  • Leandro Fonseca Prudente
  • D. R. Souza
    • Categories Multi-Criteria Optimization, Unconstrained Optimization Tags , , , ,

    We propose a BFGS method with Wolfe line searches for unconstrained multiobjective optimization problems. The algorithm is well defined even for general nonconvex problems. Global convergence and R-linear convergence to a Pareto optimal point are established for strongly convex problems. In the local convergence analysis, if the objective functions are locally strongly convex with Lipschitz … Read more

    On the exactness of the eps-constraint method for bi-objective integer nonlinear programming

  • Marianna De Santis
  • Daniele Patria
    • Categories (Mixed) Integer Nonlinear Programming, Multi-Criteria Optimization Tags , , ,

    The eps-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with problems having two nonlinear objective functions and integrality constraints on the variables. Under specific assumptions, we prove that the number of … Read more

    A Vectorization Scheme for Nonconvex Set Optimization Problems

  • Gabriele Eichfelder
  • Ernest Quintana
  • Stefan Rocktäschel
    • Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization Tags , , , ,

    In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from multiobjective optimization. Our strategy consists of deriving a parametric family of multiobjective optimization problems whose optimal solution … Read more

    A Multiobjective Approach for Sector Duration Optimization in Stereotactic Radiosurgery Treatment Planning

  • Oylum Seker
  • Mucahit Cevik
  • Merve Bodur
  • Young Lee-Bartlett
  • Mark Ruschin
    • Categories Biomedical Applications, Multi-Criteria Optimization Tags , , , ,

    Sector duration optimization (SDO) is a problem arising in treatment planning for stereotactic radiosurgery on Gamma Knife. Given a set of isocenter locations, SDO aims to select collimator size configurations and irradiation times thereof such that target tissues receive prescribed doses in a reasonable amount of treatment time, while healthy tissues nearby are spared. We … Read more

    Solving set-valued optimization problems using a multiobjective approach

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

    Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimzation problems even for specific cases. In this paper we study set-valued optimization problems and develop a multiobjective optimization … Read more

    Twenty years of continuous multiobjective optimization in the twenty-first century

  • Gabriele Eichfelder
    • Categories Global Optimization, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    The survey highlights some of the research topics which have attracted attention in the last two decades within the area of mathematical optimization of multiple objective functions. We give insights into topics where a huge progress can be seen within the last years. We give short introductions to the specific sub-fields as well as some … Read more