Optimization over the Pareto Outcome set associated with a Convex Bi-Objective Optimization Problem: Theoretical Results, Deterministic Algorithm and Application to the Stochastic case

  • Henri Bonnel
  • Julien Collonge
    • Categories Multi-Criteria Optimization, Stochastic Programming

    Our paper consists of two main parts. In the first one, we deal with the deterministic problem of minimizing a real valued function $f$ over the Pareto set associated with a deterministic convex bi-objective optimization problem (BOP), in the particular case where $f$ depends on the objectives of (BOP), i.e. we optimize over the Pareto … Read more

    Variational Analysis of Circular Cone Programs

  • Jein-Shan Chen
  • Boris S. Mordukhovich
  • Jinchuan Zhou
    • Categories Linear, Cone and Semidefinite Programming, Nonsmooth Optimization Tags , , , , , , ,

    This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming important for optimization theory and its applications. First we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/coderivatives of the projection operator associated with the circular cone. Then we … Read more

    Improving direct search algorithms by multilevel optimization techniques

  • Emanuele Frandi
  • Alessandra Papini
    • Categories Nonlinear Optimization

    Direct Search algorithms are classical derivative-free methods for optimization. Though endowed with solid theoretical properties, they are not well suited for large-scale problems due to slow convergence and scaling issues. In this paper, we discuss how such limitations can be circumvented, on problems for which a hierarchy of objective functions is available, by using multilevel … Read more

    A Note on Lerner Index, Cross-Elasticity and Revenue Optimization Invariants

  • Alexander Kushkuley
  • Su-Ming Wu
    • Categories Applications - OR and Management Sciences, Nonlinear Optimization Tags , , , ,

    We study common properties of retail pricing models in a general framework of calculus of variations. In particular, we observe that for any demand model, optimal de-seasoned revenue rate divided by price elasticity is time invariant. We also obtain a generalization of a well known inverse relationship between price elasticity of demand and Lerner index. … Read more

    A Stochastic Quasi-Newton Method for Large-Scale Optimization

  • Richard H. Byrd
  • Samantha Hansen
  • Jorge Nocedal
  • Yoram Singer
    • Categories Data-Mining, Stochastic Programming, Unconstrained Optimization Tags , , ,

    Abstract The question of how to incorporate curvature information in stochastic approximation methods is challenging. The direct application of classical quasi- Newton updating techniques for deterministic optimization leads to noisy curvature estimates that have harmful effects on the robustness of the iteration. In this paper, we propose a stochastic quasi-Newton method that is efficient, robust … Read more

    Alternating projections and coupling slope

  • Dmitriy Drusvyatskiy
  • Adrian Lewis
  • Alexander D. Ioffe
    • Categories Nonsmooth Optimization, Other Topics Tags , , , ,

    We consider the method of alternating projections for finding a point in the intersection of two possibly nonconvex closed sets. We present a local linear convergence result that makes no regularity assumptions on either set (unlike previous results), while at the same time weakening standard transversal intersection assumptions. The proof grows out of a study … Read more

    Mitigating Uncertainty via Compromise Decisions in Two-stage Stochastic Linear Programming

  • Suvrajeet Sen
  • Yifan Liu
    • Categories Stochastic Programming Tags , ,

    Stochastic Programming (SP) has long been considered as a well-justified yet computationally challenging paradigm for practical applications. Computational studies in the literature often involve approximating a large number of scenarios by using a small number of scenarios to be processed via deterministic solvers, or running Sample Average Approximation on some genre of high performance machines … Read more

    A Block Successive Upper Bound Minimization Method of Multipliers for Linearly Constrained Convex Optimization

  • Tsung-Hui Chang
  • Zhi-Quan Luo
  • Shiqian Ma
  • Mingyi Hong
  • Meisam Razaviyayn
  • Xiangfeng Wang
    • Categories Convex Optimization, Nonsmooth Optimization Tags , ,

    Consider the problem of minimizing the sum of a smooth convex function and a separable nonsmooth convex function subject to linear coupling constraints. Problems of this form arise in many contemporary applications including signal processing, wireless networking and smart grid provisioning. Motivated by the huge size of these applications, we propose a new class of … Read more

    Distributed Optimization Methods for Large Scale Optimal Control

  • Attila Kozma
    • Categories Optimization of Simulated Systems, Quadratic Programming Tags , , , , ,

    This thesis aims to develop and implement both nonlinear and linear distributed optimization methods that are applicable, but not restricted to the optimal control of distributed systems. Such systems are typically large scale, thus the well-established centralized solution strategies may be computationally overly expensive or impossible and the application of alternative control algorithms becomes necessary. … Read more

    On QPCCs, QCQPs and Completely Positive Programs

  • Lijie Bai
  • John E. Mitchell
  • Jong-Shi Pang
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming

    This paper studies several classes of nonconvex optimization problems defined over convex cones, establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs. The problems being studied include: a quadratically constrained quadratic program (QCQP), a quadratic program with complementarity constraints (QPCC), and rank constrained semidefinite programs. Our results … Read more