Nonlinear Optimization

Efficient Warm-Start Strategies for Nash-based Linear Complementarity Problems via Bilinear Approximation

  • Dominic C. Flocco
  • Steven A. Gabriel
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Global Optimization Applications Tags , , ,

    We present an effective warm-starting scheme for solving large linear complementarity problems (LCPs) arising from Nash equilibrium problems. The approach generates high-quality starting points that, when passed to the PATH solver, yield substantial reductions in computational time and variance. Our warm-start routine reformulates each agent’s LP using strong duality, leading to a master problem with … Read more

    Modeling Binary Relations in Piecewise-Linear Approximations

  • Kristin Braun
  • Robert Burlacu
  • Tobias Kuen
  • kristinarolsing
    • Categories Combinatorial Optimization, Global Optimization Theory, Nonlinear Optimization Tags , , , , ,

    Over the last decades, using piecewise-linear mixed-integer relaxations of nonlinear expressions has become a strong alternative to spatial branching for solving mixed-integer nonlinear programs. Since these relaxations give rise to large numbers of binary variables that encode interval selections, strengthening them is crucial. We investigate how to exploit the resulting combinatorial structure by integrating cutting-plane … Read more

    A Projected Stochastic Gradient Method for Finite-Sum Problems with Linear Equality Constraints

  • Nataša Krklec Jerinkić
  • Benedetta Morini
  • Mahsa Yousefi
    • Categories Nonlinear Optimization, Optimization in Data Science Tags , ,

    A stochastic gradient method for finite-sum minimization subject to deterministic linear constraints is proposed and analyzed. The procedure presented adapts the projected gradient method on a convex set to the use of both a stochastic gradient and a possibly inexact projection map. Under standard assumptions in the field of stochastic gradient methods, we provide theoretical … Read more

    Improved Analysis of Restarted Accelerated Gradient and Augmented Lagrangian Methods via Inexact Proximal Point Frameworks

  • Matthew X. Burns
  • Jiaming Liang
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Convex Optimization Tags , , , ,

    This paper studies a class of double-loop (inner-outer) algorithms for convex composite optimization. For unconstrained problems, we develop a restarted accelerated composite gradient method that attains the optimal first-order complexity in both the convex and strongly convex settings. For linearly constrained problems, we introduce inexact augmented Lagrangian methods, including a basic method and an outer-accelerated … Read more

    Modelling and Analysis of an Inverse Parameter Identification Problem in Piezoelectricity

  • Raphael Kuess
  • Daniel Walter
  • Andrea Walther
    • Categories Infinite Dimensional Optimization, Nonlinear Systems and Least-Squares, Systems governed by Differential Equations Optimization Tags , , , , ,

    Piezoelectric material behavior is mathematically described by coupled hyperbolic-elliptic partial differential equations (PDEs) governing mechanical displacement and electrical potential. This paper presents advancements in the theory of identifying material parameters in piezoelectric PDEs. We focus on modeling and analyzing the inverse problem assuming matrix-valued Sobolev-Bochner parameters to encompass a time and space-dependent setting and thus … Read more

    A General Penalty-Method and a General Regularization-Method for Cardinality-Constrained Optimization Problems

  • Felix P. Broesamle
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    We consider cardinality-constrained optimization problems (CCOPs), which are general nonlinear programs with an additional constraint limiting the number of nonzero continuous variables. The continuous reformulation of CCOPs involves complementarity constraints, which pose significant theoretical and computational challenges. To address these difficulties, we propose and analyze two numerical solution approaches: a general penalty method and a … Read more

    Fast Presolving Framework For Sparsity Constrained Convex Quadratic Programming: Screening-Based Cut Generation and Selection

  • Haozhe Tan
  • Guanyi Wang
    • Categories (Mixed) Integer Nonlinear Programming, Cutting Plane Approaches, Nonlinear Optimization Tags , , , ,

    Screening is widely utilized for Mixed-Integer Programming (MIP) presolving. It aims to certify a priori whether one or multiple specific binary variables can be fixed to optimal values based on solutions from convex relaxations. This paper studies the challenge of solving Sparsity-constrained (strongly) Convex Quadratic Programming (SCQP) and proposes the Screening-based Cut Presolving Framework (SCPF). … Read more

    Curvature-oriented variance reduction methods for nonconvex stochastic optimization

  • Qiankun Shi
  • Shiji Zuo
  • liuyx
  • Xiao Wang
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    When pursuing an approximate second-order stationary point in nonconvex constrained stochastic optimization, is it possible to design a stochastic second-order method that achieves the same sample complexity order as in the unconstrained setting? To address this question in this paper, we first introduce Carme, a curvature-oriented variance reduction method designed for unconstrained nonconvex stochastic optimization. … Read more

    Voronoi Conditional Gradient Method for Constrained Nonconvex Optimization

  • Abbas Khademi
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    The Conditional Gradient method offers a computationally efficient, projection-free framework for constrained problems; however, in nonconvex settings it may converge to stationary points of low quality. We propose the Voronoi Conditional Gradient (VCG) method, a geometric heuristic that systematically explores the feasible region by constructing adaptive Voronoi partitions from previously discovered stationary points. VCG incrementally … Read more

    Sensitivity-informed identification of temperature-dependent piezoelectric material parameters

  • Raphael Kuess
  • Olga Friesen
  • Leander Claes
  • Andrea Walther
    • Categories Applications - Science and Engineering, Control Applications, Nonlinear Optimization Tags , , ,

    An accurate characterization of temperature-dependent material parameters of piezoceramics is crucial for the design and simulation of reliable sensors and actuators. This characterization is typically formulated as an ill-posed inverse problem, which is challenging to solve not only because of its ill-posedness, but also because of parameter sensitivities, which vary by several orders of magnitude … Read more