Nonlinear Optimization

An objective-function-free algorithm for nonconvex stochastic optimization with deterministic equality and inequality constraints

  • Serge Gratton
  • Philippe L. Toint
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , ,

    An algorithm is proposed for solving optimization problems with stochastic objective and deterministic equality and inequality constraints. This algorithm is objective-function-free in the sense that it only uses the objective’s gradient and never evaluates the function value. It is based on an adaptive selection of function-decreasing and constraint-improving iterations, the first ones using an Adagrad-type … Read more

    Beyond binarity: Semidefinite programming for ternary quadratic problems

  • Frank de Meijer
  • Veronica Piccialli
  • Renata Sotirov
  • Antonio M. Sudoso
    • Categories (Mixed) Integer Nonlinear Programming, Quadratic Programming, Semi-definite Programming Tags , , ,

    We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in {0,±1}. While semidefinite programming (SDP) techniques are well established for {0,1}- and {±1}-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation … Read more

    On lifting strategies for optimal control problems

  • Robert Lampel
  • Sebastian Sager
    • Categories Nonlinear Optimization Tags , , , ,

    The representation of a function in a higher-dimensional space, often referred to as lifting, can be used to reduce complexity. We investigate how lifting affects the convergence properties of Newton-type methods. For the first time, we conduct a systematic comparison of several lifting strategies on a set of 40 optimal control problems. In addition, we … Read more

    Zeroth-Order Methods for Nonconvex-Strongly Concave Stochastic Minimax Problems with Decision-Dependent Distributions

  • Zili Luo
  • Yongchao Liu
    • Categories Nonlinear Optimization, Optimization in Data Science, Stochastic Programming Tags , , , ,

    Stochastic minimax problems with decision-dependent distributions (SMDD) have emerged as a crucial framework for modeling complex systems where data distributions drift in response to decision variables. Most existing methods for SMDD rely on an explicit functional relationship between the decision variables and the probability distribution. In this paper, we propose two sample-based zeroth-order algorithms, namely … Read more

    Bregman Regularized Proximal Point Methods for Computing Projected Solutions of Quasi-equilibrium Problems

  • Richard Osward
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonlinear Optimization Tags , , ,

    In this paper, we propose two Bregman regularized proximal point methods that provide flexibility to compute projected solutions for quasi-equilibrium problems. Each method has one Bregman projection onto the feasible set and the regularized equilibrium problem. Under standard assumptions, we prove that the methods are well-defined and that the sequences they generate converge to a … Read more

    Preconditioned Proximal Gradient Methods with Conjugate Momentum: A Subspace Perspective

  • Jian Chen
  • Xinmin Yang
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    In this paper, we propose a descent method for composite optimization problems with linear operators. Specifically, we first design a structure-exploiting preconditioner tailored to the linear operator so that the resulting preconditioned proximal subproblem admits a closed-form solution through its dual formulation. However, such a structure-driven preconditioner may be poorly aligned with the local curvature … Read more

    Strong convergence, perturbation resilience and superiorization of Generalized Modular String-Averaging with infinitely many input operators

  • Kay Barshad
  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , , ,

    We study the strong convergence and bounded perturbation resilience of iterative algorithms based on the Generalized Modular String-Averaging (GMSA) procedure for infinite sequences of input operators under a general admissible control. These methods address a variety of feasibility-seeking problems in real Hilbert spaces, including the common fixed point problem and the convex feasibility problem. In … Read more

    Solving Chance Constrained Programs via a Penalty based Difference of Convex Approach

  • Zhiping Li
  • Nan Jiang
  • Rujun Jiang
    • Categories Nonlinear Optimization, Stochastic Programming Tags , , ,

    We develop two penalty based difference of convex (DC) algorithms for solving chance constrained programs. First, leveraging a rank-based DC decomposition of the chance constraint, we propose a proximal penalty based DC algorithm in the primal space that does not require a feasible initialization. Second, to improve numerical stability in the general nonlinear settings, we … Read more

    A Successive Proximal DC Penalty Method with an Application to Mathematical Programs with Complementarity Constraints

  • Dominic C. Flocco
  • Steven A. Gabriel
  • Trine Krogh Boomsma
  • Martin Schmidt
  • Miguel A. Lejeune
    • Categories Bilevel Optimization, Constrained Nonlinear Optimization, Global Optimization Tags , , , ,

    We develop a successive, proximal difference-of-convex (DC) function penalty method for solving DC programs with DC constraints. The proposed approach relies on a DC penalty function that measures the violation of constraints and leads to a penalty reformulation sharing the same solution set as the original problem. The resulting penalty problem is a DC program … Read more

    On the Complexity of Subgradient Methods for Trilevel Hierarchical Generalized Variational Inequalities

  • Lorenzo Lampariello
  • Simone Sagratella
  • Valerio Giuseppe Sasso
    • Categories Convex Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    We study generalized variational inequalities with a three-level hierarchical structure. This setting extends nested GVI models beyond the bilevel case, for which $\mathcal{O}(\delta^{-4})$ complexity bounds are known for any prescribed positive tolerance $\delta$, to a fully three-level hierarchical structure. We analyze a projected averaged subgradient method combined with a Tikhonov-like regularization scheme. Under compactness, maximal … Read more