A Globally Linearly Convergent Method for Pointwise Quadratically Supportable Convex-Concave Saddle Point Problems

A Primal-Dual Algorithm with Line Search for General Convex-Concave Saddle Point Problems

In this paper, we propose a primal-dual algorithm with a novel momentum term using the partial gradients of the coupling function that can be viewed as a generalization of the method proposed by Chambolle and Pock in 2016 to solve saddle point problems defined by a convex-concave function $\mathcal L(x,y)=f(x)+Φ(x,y)-h(y)$ with a general coupling term $Φ(x,y)$ that is not assumed to be bilinear. Assuming $\nabla_xΦ(\cdot,y)$ is Lipschitz for any fixed $y$, and $\nabla_yΦ(\cdot,\cdot)$ is Lipschitz, we show that the iterate sequence converges to a saddle point; and for any $(x,y)$, we derive er…

二次正则化最优传输的样本复杂度

Sample Complexity of Quadratically Regularized Optimal Transport

针对最优传输的维度灾难问题，本文证明二次正则化最优传输（QOT）具有参数化样本复杂度，通过分析其耦合支撑集正则性并建立中心极限定理，解决了QOT因势不光滑被认为存在维度灾难的问题。

Synthesis in pMDPs: A Tale of 1001 Parameters

This paper considers parametric Markov decision processes (pMDPs) whose transitions are equipped with affine functions over a finite set of parameters. The synthesis problem is to find a parameter valuation such that the instantiated pMDP satisfies a specification under all strategies. We show that this problem can be formulated as a quadratically-constrained quadratic program (QCQP) and is non-convex in general. To deal with the NP-hardness of such problems, we exploit a convex-concave procedure (CCP) to iteratively obtain local optima. An appropriate interplay between CCP solvers and probabi…

求解非凸最小化问题的规范原始-对偶方法

Canonical Primal-Dual Method for Solving Non-convex Minimization Problems

提出一种基于规范对偶理论的新原始-对偶算法，将非凸最小化问题重构为凸-凹鞍点优化问题，并通过二次扰动原始-对偶方法求解，数值实验表明性能优于现有方法。

Adaptive Primal-Dual Stochastic Gradient Method for Expectation-constrained Convex Stochastic Programs

Stochastic gradient methods (SGMs) have been widely used for solving stochastic optimization problems. A majority of existing works assume no constraints or easy-to-project constraints. In this paper, we consider convex stochastic optimization problems with expectation constraints. For these problems, it is often extremely expensive to perform projection onto the feasible set. Several SGMs in the literature can be applied to solve the expectation-constrained stochastic problems. We propose a novel primal-dual type SGM based on the Lagrangian function. Different from existing methods, our metho…

Robust Policy Synthesis for Uncertain POMDPs via Convex Optimization

We study the problem of policy synthesis for uncertain partially observable Markov decision processes (uPOMDPs). The transition probability function of uPOMDPs is only known to belong to a so-called uncertainty set, for instance in the form of probability intervals. Such a model arises when, for example, an agent operates under information limitation due to imperfect knowledge about the accuracy of its sensors. The goal is to compute a policy for the agent that is robust against all possible probability distributions within the uncertainty set. In particular, we are interested in a policy that…

具有自适应步长的重启加速原始-对偶算法用于非线性锥约束凸优化

Restarted Accelerated Primal-Dual Algorithms with Adaptive Stepsizes for Nonlinear Conic Constrained Convex Optimization

针对非线性锥约束凸优化问题（含二次约束二次规划），提出带回溯的自适应步长重启加速原始-对偶算法，利用一阶信息和矩阵-向量乘积，在KKT映射度量次正则性下证明全局线性收敛，数值实验显示非单调自适应步长和GPU加速效果显著。

Convex Optimization for Parameter Synthesis in MDPs

Probabilistic model checking aims to prove whether a Markov decision process (MDP) satisfies a temporal logic specification. The underlying methods rely on an often unrealistic assumption that the MDP is precisely known. Consequently, parametric MDPs (pMDPs) extend MDPs with transition probabilities that are functions over unspecified parameters. The parameter synthesis problem is to compute an instantiation of these unspecified parameters such that the resulting MDP satisfies the temporal logic specification. We formulate the parameter synthesis problem as a quadratically constrained quadrati…