(Communicated by Liqun Qi)

(Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday)

Abstract: Inspired by the recent work of Shapiro et al., we propose a Bayesian distributionally robust Nash equilibrium (BDRNE) model where each player lacks complete information on the true probability distribution of the underlying exogenous uncertainty represented by a random variable and subsequently determines the optimal decision by solving a Bayesian distributionally robust optimization (BDRO) problem under the Nash conjecture. Unlike most of the DRO models in the literature, the BDRO model assumes (a) the true unknown distribution of the random varialbe can be approximated by a randomized parametric family of distributions, (b) the average of the worst-case expected value of the objective function with respect to the posterior distribution of the parameter, instead of the worst-case expected value of the objective function is considered in each player’s decision making, and (c) the posterior distribution of the parameter is updated as more and more sampling information of the random variable is gathered. Under some moderate conditions, we demonstrate the existence of a BDRNE and derive asymptotic convergence of the equilibrium as the sample size increases. Moreover, we propose to solve the BDRNE problem by Gauss-Seidel-type iterative method in the case when the ambiguity set of each player is constructed via Kullback-Leibler (KL) divergence. Finally, we apply the BDRNE model to a price competition problem under multinomial logit demand. The preliminary numerical test results show that the proposed model and computational scheme perform well.

(Communicated by Liqun Qi)

Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday

Abstract: Mean inequalities on the second order cone have been studied as an extension of the SPD cone. In this article, we turn our eyes on non-symmetric cones. In fact, we investigate two types of decompositions associated with circular cones, and establish their own mean inequalities. These inequalities are ground bricks for further study regarding circular cone optimization. We also find under the condition $0<\theta<\frac{\pi}{4}$ some inequalities cannot hold if we apply different decomposition, and correspondingly we raise a conjecture.

(Communicated by Liqun Qi)

(Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday)

Abstract: In this work using subgradient algorithm we study a minimization problem with a convex objective function on a domain, which is the solution set of a common fixed point problem with a countable family of quasi-nonexpansive mappings. Our algorithm generates a sequence of iterates which are approximate solutions of the corresponding fixed point problem. Additionally, also either this sequence converges to a solution of our minimization problem or the sequence is strictly Fejer monotone regarding the solution set of the common fixed point problem.

(Communicated by Liqun Qi)

(Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday)

Abstract: Existing decentralized algorithms usually require knowledge of problem parameters for updating local iterates. For example, the hyperparameters (such as learning rate) usually require the knowledge of Lipschitz constant of the global gradient or topological information of the communication networks, which are usually not accessible in practice. In this paper, we propose D-NASA, the first algorithm for decentralized nonconvex stochastic optimization that requires no prior knowledge of any problem parameters. We show that D-NASA has the optimal rate of convergence for nonconvex objectives under very mild conditions and enjoys the linear-speedup effect, i.e. the computation becomes faster as the number of nodes in the system increases. Extensive numerical experiments are conducted to support our findings.

(Communicated by Liqun Qi)

(Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday)

    Convex separable quadratic optimization problems arise in numerous practical applications. These problems take the form:

$\begin{align*}\min _{\boldsymbol{y} \in \mathbb{R}^n} f(\boldsymbol{y}) & =\boldsymbol{y}^T \Delta \boldsymbol{y}+\boldsymbol{\alpha}^T \boldsymbol{y} \\ \text { s.t. } g_{i}(\boldsymbol{y})& = \boldsymbol{y}^T \Theta_i \boldsymbol{y} +\boldsymbol{\beta}_i ^T \boldsymbol{y} +\sigma_i \leq 0,~ i=1,\dots,m, \label{SQPQC}\tag{SQPQC}\\\boldsymbol{y} & \in[\boldsymbol{l}, \boldsymbol{u}],\end{align*}$

where $\Delta$ and $\Theta_i$ are $n \times n$ diagonal matrices.

    Building on an iterative resolution scheme for the KKT system, we develop an efficient method for solving problem \ref{SQPQC}. We begin by applying the dual method for problem (SQPQC) with one single quadratic constraint, exploiting some distinct properties of this subclass, such as the differentiability of the dual function and the uniqueness of the optimal solution of the dual problem. Leveraging on the properties, we then develop a highly efficient algorithm for the general case of multiple quadratic constraints, and we show that the proposed approach leads to a dual coordinate ascent algorithm. Finally, we prove the convergence of the proposed algorithm and demonstrate, through extensive numerical experiments, that it outperforms the Gurobi solver, particularly for large-scale convex separable quadratic programming problems.

        In this paper, an optimal control approach is proposed to compute Newton's direction in $O(n)$ flops for certain large scale optimization problems, where $n$ is the number of variables. Consequently, these optimization problems can be solved in $O(n)$ flops by two optimal control algorithms (including Newton's method) which have local quadratic convergence. The essence of our methodology is to convert the unconstrained optimization problem into a discrete-time optimal control problem. Our discussion begins with the introduction of a new framework to describe and identify the structure of a general function with a large number of variables. Based on the new framework, a sufficient condition is established so that for any large scale unconstrained optimization problem satisfying this sufficient condition, its Newton's direction can be computed in $O(n)$ flops. Our application of this sufficient condition on a collection of large scale unconstrained optimization problems in the literature indicates that the majority of these problems are amenable to this approach.

(Communicated by Lingchen Kong)

This paper is dedicated to Professor Terry Rockafellar in celebration of his 90th birthday

     Multiobjective optimization is an important field of research closely related to applications, including statistics, engineering, management science, finance, and others. In this paper, we consider the following multiobjective optimization problem:

(Communicated by Liqun Qi)

Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday

    Most numerical methods developed for solving nonlinear programming problems are designed to find points that satisfy certain optimality conditions. While the Karush-Kuhn-Tucker conditions are well-known, they become invalid when constraint qualifications (CQ) are not met. Recent advances in sequential optimality conditions address this limitation in both first- and second-order cases, providing genuine optimality guarantees at local optima, even when CQs do not hold. However, some second-order sequential optimality conditions still require some restrictive conditions on constraints in the recent literature.

    In this paper, we propose a new strong second-order sequential optimality condition without CQs. We also show that a penalty-type method and an augmented Lagrangian method generate points satisfying these new optimality conditions.

(Communicated by Liqun Qi)

(Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday)

    We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [1] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Łojasiewicz (KŁ) properties of suitable potential functions. We also study how the KŁ exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [1], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [2,Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.

(Communicated by Michel Théra )

(Dedicated to Professor R. Tyrrell Rockafellar on the occasion of his 90th birthday)

    This work presents two optimization methods to compute, subject to constraints, a Wasserstein barycenter (WB) of finitely many empirical probability measures.

   The new measure, denoted by constrained Wasserstein barycenter, extends the applicability of the standard WB to pre-required geometrical or statistical constraints. Our first approach is an extension of the Method of Averaged Marginals (Mimouni et al., 2024) to compute WBs subject to convex constraints. In the nonconvex setting, we propose an optimization model whose necessary optimality conditions are written as a linkage problem with non-elicitable monotonicity. To solve such a linkage problem, we combine the Progressive Decoupling Algorithm (Rockafellar, 2019) with Difference-of-Convex programming techniques. We give the mathematical properties of our approaches and evaluate their numerical performances in two applications, demonstrating both their computational efficiency and the practical relevance of constrained Wasserstein barycenters.