In this paper, we present a proximal gradient method with a new non-monotone line search for solving non-convex and non-smooth composite optimization problems. The method introduces a random term to determine whether the algorithm directly accepts the Barzilai-Borwein stepsize. Our approach uses a line-form backtracking line search, which ensures that the proximal operator is computed only once per iteration, offering an advantage over existing mainstream proximal gradient methods. Under mild conditions that do not require gradient Lipschitz continuity, we establish the subsequence convergence of the proposed method. Preliminary experiments show that our method outperforms existing non-monotone line search methods in terms of performance and efficiency.
This paper studies the structure of the core in many-to-one assignment games, where firms with limited capacity hire workers in a transferable utility framework. While it is well-established that the core of these games is always non-empty and contains firm-optimal and worker-optimal outcomes, relatively little is known about its complete geometric structure. In particular, regarding the characterization and enumeration of its extreme points. In this paper, we provide a graph-theoretic criterion for core vertices: a salary vector is a vertex of the core if and only if the base graph of its associated tight digraph is connected. We also provide a necessary and sufficient condition for each side-optimal allocation in terms of the tight digraph. Based on this characterization, we develop a lexicographic procedure that generates all core vertices as they are supported by a max-min salary vector, where workers sequentially optimize their payoffs with an indication of whether the worker in this position maximizes or minimizes his/her salary.
In this paper, we discuss a class of nonconvex-nonsmooth multi-block optimization with equality and closed convex set constraints, and the aim is to propose a novel fully symmetric Peaceman–Rachford (PR) regularization hybrid distributed method (HDM). The method is based on the ideas of the PR splitting and sequential quadratic programming (SQP) methods. It incorporates the Armijo line search technique and a sequential updating scheme for the Lagrange multiplier (dual variable). To ensure the solvability of nonsmooth subproblems, we add a proximal regularization term to each of them. This guarantees the existence and uniqueness of solutions, while improving numerical stability. For smooth subproblems, we use their quadratic approximations to enhance efficiency. The search direction corresponding to the smooth sub-objective is generated by solving a small-scale quadratic programming (QP), and then takes the augmented Lagrangian function (ALF) as the merit function to perform the Armijo line search in the sequential order. As a result, the new iteration of the associated primal block variable is yielded, respectively. Once a primal block variable is updated, the method immediately updates the dual variable with the latest iteration information, called it a fully symmetric Peaceman–Rachford distributed method in this sense. Under mild regularity assumptions, the global convergence and iteration complexity are established, and then the Maratos effect can be also overcome. Furthermore, by additionally imposing the condition that the merit function satisfies the Kurdyka–Łojasiewicz (KL) property and the gradients of the smooth block-objectives are locally Lipschitz continuous, the strong convergence and the convergence rate of the proposed method are established. A large number of compared numerical experiments on medium-to-large-scale problems show the effectiveness and robustness of the proposed method.
Extending Bayesian optimization to solve high-dimensional optimization problems is a meaningful but challenging task. As the dimension of the problem increases, it becomes exponentially more difficult to find the global optimum of the acquisition function, which often worsens the performance of Bayesian optimization in high-dimensional spaces. In this work, we propose a simple and efficient approach to tackle this problem. Instead of optimizing the acquisition function in the original high-dimensional space, we propose to use the block coordinate decent approach to optimize it block by block. The block size is initialized as the dimension of the problem, and deceases adaptively according to whether a better solution is found. The adaptive approach drives the search from global exploration to local exploitation gradually. In addition, we extend our approach to parallel evaluations in this work to further accelerate the Bayesian optimization process on high-dimensional problems. Extensive experiments have been conducted to verify the effectiveness of our proposed block coordinate decent approach. The results show our proposed algorithm can significantly improve the performance of Bayesian optimization on high-dimensional problems. Also, our algorithm shows very competitive optimization efficiency compared with six state-of-the-art high-dimensional Bayesian optimization algorithms. The codes of our proposed algorithm are available at https://github.com/zhandawei/Block_Coordinate_Descent_Bayesian_Optimization.
A Kantorovich-like optimal transport problem is formulated by seeking to minimize the Choquet integral of a given cost function with respect to the α -mixture of joint belief functions with given marginals, and their dual plausibility functions. This extension of the classical probabilistic problem allows to deal with misspecified marginal distributions by relying on a parameter α∈ [0,1] that acts like a pessimism index. We show that the particular subcase given by a marginal belief function and a marginal probability measure can be used to model a game under ambiguity, through the definition of the Stackelberg-Cournot-Nash equilibrium with Dempster-Shafer uncertainty and α -maxmin preferences. An algorithm is provided for approximating an equilibrium based on a suitable entropic formulation of the defined optimal transport problem. Finally, a distinguished application is found in a market with a finite set of types of investors having misspecified distribution. Investors’ choices result in a purchase portfolio on a finite set of assets which, in turn, depends on asset prices fixed by the market maker and the interactions among investors.
