Linear, Cone and Semidefinite Programming

A New Use of Douglas-Rachford Splitting and ADMM for Identifying Infeasible, Unbounded, and Pathological Conic Programs

  • Yanli Liu
  • Ernest K. Ryu
  • Wotao Yin
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , ,

    In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas-Rachford splitting, or equivalently ADMM. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas-Rachford splitting diverge.Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly … Read more

    Size Matters: Cardinality-Constrained Clustering and Outlier Detection via Conic Optimization

  • Daniel Kuhn
  • Napat Rujeerapaiboon
  • Kilian Schindler
  • Wolfram Wiesemann
    • Categories Approximation Algorithms, Data-Mining, Linear, Cone and Semidefinite Programming Tags , , ,

    Plain vanilla K-means clustering is prone to produce unbalanced clusters and suffers from outlier sensitivity. To mitigate both shortcomings, we formulate a joint outlier-detection and clustering problem, which assigns a prescribed number of datapoints to an auxiliary outlier cluster and performs cardinality-constrained K-means clustering on the residual dataset. We cast this problem as a mixed-integer … Read more

    Lower Bound On the Computational Complexity of Discounted Markov Decision Problems

  • Yichen Chen
  • Mengdi Wang
    • Categories Dynamic Programming, Linear Programming Tags ,

    We study the computational complexity of the infinite-horizon discounted-reward Markov Decision Problem (MDP) with a finite state space $\cS$ and a finite action space $\cA$. We show that any randomized algorithm needs a running time at least $\Omega(\carS^2\carA)$ to compute an $\epsilon$-optimal policy with high probability. We consider two variants of the MDP where the … Read more

    Bad semidefinite programs with short proofs, and the closedness of the linear image of the semidefinite cone

  • Gabor Pataki
    • Categories Convex Optimization, Nonlinear Optimization, Semi-definite Programming

    Semidefinite programs (SDPs) — some of the most useful and pervasive optimization problems of the last few decades — often behave pathologically: the optimal values of the primal and dual problems may differ and may not be attained. Such SDPs are theoretically interesting and often impossible to solve. Yet, the pathological SDPs in the literature … Read more

    Exact augmented Lagrangian functions for nonlinear semidefinite programming

  • Ellen H. Fukuda
  • Bruno F. Lourenço
    • Categories Constrained Nonlinear Optimization, Semi-definite Programming Tags , ,

    In this paper, we study augmented Lagrangian functions for nonlinear semidefinite programming (NSDP) problems with exactness properties. The term exact is used in the sense that the penalty parameter can be taken appropriately, so a single minimization of the augmented Lagrangian recovers a solution of the original problem. This leads to reformulations of NSDP problems … Read more

    Polynomial Norms

  • Amir Ali Ahmadi
  • Etienne de Klerk
  • Georgina Hall
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Semi-definite Programming Tags , , ,

    In this paper, we study polynomial norms, i.e. norms that are the dth root of a degree-d homogeneous polynomial f. We first show that a necessary and sufficient condition for f^(1/d) to be a norm is for f to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from dth … Read more

    The Many Faces of Degeneracy in Conic Optimization

  • Dmitriy Drusvyatskiy
  • Henry Wolkowicz
    • Categories Combinatorial Optimization, Convex Optimization, Semi-definite Programming Tags , , , , ,

    Slater’s condition — existence of a “strictly feasible solution” — is a common assumption in conic optimization. Without strict feasibility, first-order optimality conditions may be meaningless, the dual problem may yield little information about the primal, and small changes in the data may render the problem infeasible. Hence, failure of strict feasibility can negatively impact … Read more

    Warm-start of interior point methods for second order cone optimization via rounding over optimal Jordan frames

  • Sertalp B. Çay
  • Imre Pólik
  • Tamás Terlaky
    • Categories Branch and Cut Algorithms, Second-Order Cone Programming Tags , , , ,

    Interior point methods (IPM) are the most popular approaches to solve Second Order Cone Optimization (SOCO) problems, due to their theoretical polynomial complexity and practical performance. In this paper, we present a warm-start method for primal-dual IPMs to reduce the number of IPM steps needed to solve SOCO problems that appear in a Branch and … Read more

    Facially dual complete (nice) cones and lexicographic tangents

  • Vera Roshchina
  • Levent Tunçel
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , ,

    We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex optimization. Using the well-known and very commonly used concept of tangent cones in nonlinear … Read more

    From Data to Decisions: Distributionally Robust Optimization is Optimal

  • Daniel Kuhn
  • Bart Paul Gerard Van Parys
  • Peyman Mohajerin Esfahani
    • Categories Robust Optimization, Second-Order Cone Programming, Stochastic Programming Tags , , , , ,

    We study stochastic programs where the decision-maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, … Read more