Properties of Utility Functions in Representing Preferences for Goods

会议的宗旨是为在数据驱动的运筹优化理论及应用领域里的卓越研究人员和实际工作者提供一个学术交流平台，推进国内外专家的充分合作，增强国内学科发展的动力。

Properties of Utility Functions in Representing Preferences for Goods

R. Tyrrell Rockafellar  (华盛顿大学)

In microeconomics it's important to understand when an agent might prefer one vector of goods over another.  Economists have agreed on a set of axioms for this, under which preferences can be characterized by utility functions:  a goods vector is better if the utility assigned to it is higher.  The axioms make the utility function be quasi-concave and perhaps quasi-smooth, in the sense of having smooth boundaries to its upper-level sets. Yet the utility function can be troublingly far from unique, and this is a serious impediment to the use of such preferences in optimization.

In this talk it will be explained how variational analysis is now able to identify additional preference axioms, likely acceptable to economists, which guarantee utility functions that are continuously twice differentiable. On arbitrarily large compact convex subsets of the positive orthant of goods vectors, those functions can moreover be taken to be concave, not just quasi-concave.  Utility is, in that way, unique up to affine rescaling, i.e., up to the size of the unit and the location of 0.

The Equivalence between the Progressive Hedging Algorithm and the Alternating Direction Method and Its Applications in Multistage Stochastic Programming

孙捷（科廷大学）

The progressive hedging algorithm of Rockafellar and Wets for multistage stochastic programming problems could be viewed as a two-block alternating direction method of multipliers. This correspondence brings in some useful results. In particular, it provides a new proof for the convergence of the progressive hedging algorithm with a flexibility in the selection of primal and dual step lengths and it helps in developing a new progressive hedging algorithm for solving risk averse stochastic optimization problems with cross constraints.

Alternating Subspace Newton and Iteratively Reweighted L1 Algorithms for Nonconvex Sparse Optimization

王浩  (上海科技大学)

This paper presents a novel second-order algorithm to minimize the sum of a continuously differentiable loss function and a nonsmooth, possibly nonconvex, sparse regularization function. Our method alternates between solving a reweighted l1-regularized subproblem and performing an inexact subspace Newton step. The reweighted l1-subproblem allows for efficient closed-form solutions via the soft-thresholding operator, avoiding the computational overhead of proximity operator calculations. The method keeps nonzero components sufficiently distant from zero as it approaches an optimal solution. It eventually transitions to a perturbed Newton method. We establish global convergence to a critical point and demonstrate local convergence under certain conditions. Numerical experiments show that our algorithm outperforms existing methods in both efficiency and solution quality across diverse model prediction problems.

Global optimization algorithms for solving linear fractional programming problems

申培萍 (华北水利水电大学)

The sum of linear ratios problem (SLRP) arises from numerous applications such asthe classic transportation problem, the production planning problem. The problem is first transformed into  its equivalent form (EP) via Charnes–Cooper transformation and some variables transformation. A convex relaxation problem (CRP) for (EP) is constructed to obtain a lower bound to the optimal value of (EP). In addition, a novel adjustable branching rule is proposed to offer tight lower bounds to the optimal values of (EP) over the corresponding sub-rectangles under some certain conditions. Also, a convex combination method is designed to update the upper bound for the optimal value of (EP). By continuously refining the initial rectangle and tackling a series of convex relaxation problems, the presented algorithm can find a global optimal solution to (EP). Moreover, we analyze the complexity result of the proposed algorithm. Finally, the feasibility and effectiveness of the algorithm are verified by preliminary numerical experiments.

Level Proximal Subdifferential, Variational Convexity, and Proximal Algorithms Beyond Convexity

罗洪林  (重庆师范大学)

Level proximal subdifferential was introduced by Rockafellar recently for studying proximal mappings of possibly nonconvex functions. As a powerful tool, level proximal subdifferential provides deep insights into variational analysis and optimization.  In this talk, I will introduce level proximal subdifferential systematically. We characterize variational convexity of a function by local firm nonexpansiveness of proximal mappings or local relative monotonicity of level proximal subdifferential, and use them to study local convergence of proximal gradient method and others for variationally convex functions. Variational sufficiency guarantees that proximal gradient method converges to local minimizers rather than just critical points. We also investigate the existence, single-valuedness and integration of level proximal subdifferential, and quantify pointwise Lipschitz smoothness of a function.

WoS 视角下的运筹与管理期刊

张济明  (上海大学)

通过深度挖掘Web of Science数据库，全面介绍国际运筹与管理期刊的整体发展现状，分析中国学者发文与引用情况。重点介绍《中国运筹学会会刊(英文)》的发展历程、数据库收录、投审稿流程及注意事项等。

Understanding Prices Through Ideas About Economic Equilibrium

Ralph Tyrrell Rockafellar  (华盛顿大学)

Prices are familiar in everyday life, but what determines them?  The standard explanation in economics rests on a mathematical model of buying and selling of goods by various agents.  It reflects an equilibrium state in which supply matches demand and no agent is interested in any further buying or selling.  The existence of such a state is established, but the theory has serious shortcomings.  Central among those is the lack of an economic process for reaching an equilibrium.  There's no realistic picture of a market and not even a role for money.  At best, the theory only produces relative values of goods.

This talk will begin with a discussion of such issues that have not yet been satisfactorily addressed.  It will then go on to describe a new mathematical model in which agents interact in pairs, with one buying a quantity of a single good from the other for an amount of money.  Simple conditions on the preferences and actions of the agents in that model guarantee that, through repetition, an equilibrium of prices and holding will in fact be achieved.

Recent Progresses on Online Linear Programming and Applications

叶荫宇  (斯坦福大学)

A natural optimization model that formulates many online learning and resource allocation decision-making with uncertainty is online linear programming (OLP), where the noisy constraint column vectors, along with the objective coefficients and decision variables, are revealed and decided sequentially in real time. In this talk, we review the near optimal algorithms and theories for solving this surprisingly general class of online (binary) linear programs and convex optimization problems. Then we present few recent developments of model/algorithms and applications in this area, including fast online-gradient learning and decision making, Bandit with Knapsacks setting, the Fisher Market with online pricing, Decoupling strategy of learning and decision-making, Wait-Less algorithms with mixed LP resolving and online gradient methods, Online learning and decision-making with batches. Hyper-gradient methods with online scaling/pre-conditioning for the gradient-based algorithms.

On the Recent Advancements of Control Parameterization Methods

余长君  (上海大学)

Control parameterization methods have been widely used in solving optimal control problems by discretizing control functions into a finite set of decision variables. This talk presents recent advancements in control parameterization, focusing on two key aspects. First, we explore switching time optimization, an approach that refines the timing of control switches to enhance solution quality and improve system performance. We discuss techniques for optimizing switching instants and their integration with modern numerical solvers. Second, we investigate the correlation between direct collocation and control parameterization methods, highlighting their theoretical connections and practical implications. By examining their similarities and differences, we provide insights into how these methods can be effectively applied to challenging control problems. The talk concludes with discussions on future directions and potential applications of control parameterization in emerging fields.

The Rate of Convergence of Augmented Lagrangian Method for  Minimax Optimization Problems with Semidefinite  Constraints

张立卫  ( 东北大学 )

The augmented Lagrangian method  for solving a class of minimax optimization problems with semidefinite constraints is investigated.  Firstly, we propose the second-order optimality conditions for the minimax optimization problem. Under the second-order sufficiency optimality conditions, we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters  is bounded and the convergence rate is superlinear if the sequence of penalty parameters is increasing to infinity. Finally, we use the second-order derivative of the value function to estimate the rate of convergence of the sequence of multiplier vectors.(This is a joint work with Professor Yu-Hong Dai)

主特征值的优化新视角：差模型与新算法

计算矩阵的最大特征值是一个基础性问题。经典方法主要基于瑞利商形式来求解。本报告研究了主特征值问题的差模型，基于此，经典的幂法可被视作凸差算法的一种具体形式，也可被视为步长为1/2的梯度法(优化理论表明小于1的步长都能概率1地收敛到最大特征值)。本报告基于精妙的优化策略进行设计，提出了裂-聚法。新算法无需借助谱先验知识，即可实现最大程度的加速，并且仅依靠矩阵-向量乘法运算即可完成计算。此外，我们对算法的收敛性质展开了深入探讨。通过在合成数据集和真实数据集上进行广泛的数值实验，结果表明裂-聚法相较于幂法实现了超过 10 倍的加速效果。

Aubin Property and Strong Regularity Are Equivalent for Nonlinear Second-Order Cone Programming

陈亮  (湖南大学)

This paper solves a fundamental open problem in variational analysis on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming (SOCP) at a locally optimal solution. We achieve this by introducing a reduction approach to the Aubin property characterized by the Mordukhovich criterion and a lemma of alternative choices on cones to replace the S-lemma used in Outrata and Ramírez [SIAM J. Optim., 21 (2011), pp. 789–823] and Opazo, Outrata, and Ramírez [SIAM J. Optim., 27 (2017), pp. 2143–2151], where the same SOCP was considered under the strict complementarity condition except for possibly only one block of constraints. As a byproduct, we also offer a new approach to the well-known result of Dontchev and Rockafellar [SIAM J. Optim., 6 (1996), pp. 1087–1105] on the equivalence of the two concepts in conventional nonlinear programming.