saddle point problem

A Symmetric Primal-Dual method with two extrapolation steps for Composite Convex Optimization

  • Feng Ma
    • Categories Convex Optimization Tags , , ,

    Symmetry is a recurring feature in algorithms for monotone operator theory and convex optimization, particularly in problems involving the sum of two operators, as exemplified by the Peaceman–Rachford splitting scheme. However, in more general settings—such as composite optimization problems with three convex functions or structured convex-concave saddle-point formulations—existing algorithms often exhibit inherent asymmetry. In particular, … Read more

    Primal-dual proximal bundle and conditional gradient methods for convex problems

  • Jiaming Liang
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle … Read more

    Fast convergence of the primal-dual dynamical system and algorithms for a nonsmooth bilinearly coupled saddle point problem

  • Kewei Ding
  • Jörg Fliege
  • Phan Vuong
    • Categories Game Theory, Nonsmooth Optimization Tags , , , ,

    This paper is devoted to study the convergence rates of a second-order dynamical system and its corresponding discretizations associated with a nonsmooth bilinearly coupled convex-concave saddle point problem. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on the implicit discretization, we propose a … Read more

    Doubly stochastic primal dual splitting algorithm with variance reduction for saddle point problems

  • Dimitri Papadimitriou
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    The structured saddle-point problem involving the infimal convolution in real Hilbert spaces finds applicability in many applied mathematics disciplines. For this purpose, we develop a stochastic primal-dual splitting algorithm with loopless variance-reduction for solving this generic problem. We first prove the weak almost sure convergence of the iterates. We then demonstrate that our algorithm achieves … Read more

    Fast convergence of inertial primal-dual dynamics and algorithms for a bilinearly coupled saddle point problem

  • Phan Vuong
    • Categories Convex Optimization Tags , , , , ,

    This paper is devoted to study the convergence rates of a second-order dynamical system and its corresponding discretization associated with a continuously differentiable bilinearly coupled convex-concave saddle point problem. First, we consider the second-order dynamical system with asymptotically vanishing damping term and show the existence and uniqueness of the trajectories as global twice continuously differentiable … Read more

    Generalized asymmetric forward-backward-adjoint algorithms for convex-concave saddle-point problem

  • Jianchao Bai
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , ,

    The convex-concave minimax problem, also known as the saddle-point problem, has been extensively studied from various aspects including the algorithm design, convergence condition and complexity. In this paper, we propose a generalized asymmetric forward-backward-adjoint algorithm (G-AFBA) to solve such a problem by utilizing both the proximal techniques and the extrapolation of primal-dual updates. Besides applying … Read more

    Graph topology invariant gradient and sampling complexity for decentralized and stochastic optimization

  • Guanghui Lan
  • Yuyuan Ouyang
  • Yi Zhou
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming Tags , , ,

    One fundamental problem in decentralized multi-agent optimization is the trade-off between gradient/sampling complexity and communication complexity. We propose new algorithms whose gradient and sampling complexities are graph topology invariant, while their communication complexities remain optimal. For convex smooth deterministic problems, we propose a primal dual sliding (PDS) algorithm that computes an $\epsilon$-solution with $O((\tilde{L}/\epsilon)^{1/2})$ gradient … Read more

    On the Convergence of Projected Alternating Maximization for Equitable and Optimal Transport

  • Minhui Huang
  • Shiqian Ma
  • Lifeng Lai
    • Categories Convex Optimization Tags , , , , , ,

    This paper studies the equitable and optimal transport (EOT) problem, which has many applications such as fair division problems and optimal transport with multiple agents etc. In the discrete distributions case, the EOT problem can be formulated as a linear program (LP). Since this LP is prohibitively large for general LP solvers, Scetbon \etal \cite{scetbon2021equitable} … Read more

    Equivalent second-order cone programs for distributionally robust zero-sum games

  • Ayush Agarwal
  • Abdel Lisser
  • Vikas Vikram Singh
  • Navnit Yadav
    • Categories Game Theory, Stochastic Programming Tags , , , ,

    We consider a two player zero-sum game with stochastic linear constraints. The probability distributions of the vectors associated with the constraints are partially known. The available information with respect to the distribution is based mainly on the two first moments. In this vein, we formulate the stochastic linear constraints as distributionally robust chance constraints. We … Read more

    A search direction inspired primal-dual method for saddle point problems

  • Shengjie Xu
    • Categories Convex and Nonsmooth Optimization, Convex Optimization Tags , , , , ,

    The primal-dual hybrid gradient algorithm (PDHG), which is indeed the Arrow-Hurwicz method, has been widely used in image processing areas. However, the convergence of PDHG was established only under some restrictive conditions in the literature, and it is still missing for the case without extra constraints. In this paper, from a perspective of the variational … Read more