This article addresses decentralized robust portfolio optimization based on multiagent systems. Decentralized robust portfolio optimization is first formulated as two distributed minimax optimization problems in a Markowitz return-risk framework. Cooperative-competitive multiagent systems are developed and applied for solving the formulated problems. The multiagent systems are shown to be able to reach consensuses in the expected stock prices and convergence in investment allocations through both intergroup and intragroup interactions. Experimental results of the multiagent systems with stock data from four major markets are elaborated to substantiate the efficacy of multiagent systems for decentralized robust portfolio optimization.

In this paper, we consider the so-called worst-case linear optimization (WCLO) with uncertainties on the right-hand side of the constraints. Such a problem often arises in applications such as in systemic risk estimation in finance and stochastic optimization. We first show that the WCLO problem with the uncertainty set corresponding to the [Formula: see text]p-norm ((WCLOp)) is NP-hard for p ɛ (1,∞). Second, we combine several simple optimization techniques, such as the successive convex optimization method, quadratic convex relaxation, initialization, and branch-and-bound (B&B), to develop an algorithm for (WCLO2) that can find a globally optimal solution to (WCLO2) within a prespecified ε-tolerance. We establish the global convergence of the algorithm and estimate its complexity. We also develop a finite B&B algorithm for (WCLO∞) to identify a global optimal solution to the underlying problem, and establish the finite convergence of the algorithm. Numerical experiments are reported to illustrate the effectiveness of our proposed algorithms in finding globally optimal solutions to medium and large-scale WCLO instances.

Traditional modeling on the mean-variance portfolio selection often assumes a full knowledge on statistics of assets' returns. It is, however, not always the case in real financial markets. This paper deals with an ambiguous mean-variance portfolio selection problem with a mixture model on the returns of risky assets, where the proportions of different component distributions are assumed to be unknown to the investor, but being constants (in any time instant). Taking into consideration the updates of proportions from future observations is essential to find an optimal policy with active learning feature, but makes the problem intractable when we adopt the classical methods. Using reinforcement learning, we derive an investment policy with a learning feature in a two-level framework. In the lower level, the time-decomposed approach (dynamic programming) is adopted to solve a family of scenario subcases where in each case the series of component distributions along multiple time periods is specified. At the upper level, a scenario-decomposed approach (progressive hedging algorithm) is applied in order to iteratively aggregate the scenario solutions from the lower layer based on the current knowledge on proportions, and this two-level solution framework is repeated in a manner of rolling horizon. We carry out experimental studies to illustrate the execution of our policy scheme.

In this paper, we investigate a portfolio optimization methodology using nonparametric value at risk (VaR). In particular, we adopt kernel VaR and quadratic VaR as risk measures. As the resulting models are nonconvex and nonsmooth optimization problems, albeit with some special structures, we propose some specially devised block coordinate descent (BCD) methods for finding approximate or local optimal solutions. Computational results show that the BCD methods are efficient for finding local solutions with good quality and they compare favorably with the branch-and-bound method-based global optimal solution procedures. From the simulation test and empirical analysis that we carry out, we are able to conclude that the mean-VaR models using kernel VaR and quadratic VaR are more robust compared to those using historical VaR or parametric VaR under the normal distribution assumption, especially when the information of the return distribution is limited. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0793 .

We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints that arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a “dual” method to the convexification method employing the perspective function on semicontinuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of mixed-integer quadratic programming solvers when applied to the perspective reformulations of the problem.

In this paper, we apply the game theory to study some strategic actions for retailers to fight a price war. We start by modeling a noncooperative pure pricing game among multiple competing retailers who sell a certain branded product under price-dependent stochastic demands. A unique Nash equilibrium is proven to exist under some mild conditions. We demonstrate mathematically the incentives for retailers to start a price war. Based on a strategic framework via the game theory, we illustrate the use of service level to build price walls which can prevent a huge drop in price, as well as profit. Three kinds of price walls are proposed, and the respective strengths and weaknesses have been studied. Analytical conditions, under which a price wall can effectively prevent big drops in both market share and profit, are developed. Aside from the proposed price walls, two other pricing strategies, which can lead to an all-win situation, are examined.

The newsvendor problem is a fundamental building block for inventory management with a stochastic demand. The classical newsvendor problem focuses on a sole objective of either minimizing the expected cost or maximizing the expected profit. However, the performance measure with expected value alone is insufficient, and it ignores the risk preferences of the decision makers. As a result, we carry out a mean-variance analysis of the newsvendor problem. We construct analytical models and reveal the problem's structural properties. We propose the solution schemes which help to identify the optimal solutions. Interesting findings regarding the efficient frontier, the case with a stockout penalty cost, and the safety-first objective are discussed.

In multiobjective optimization, tradeoff analysis plays an important role in determining the best search direction to reach a most preferred solution. This paper presents a new explicit interactive tradeoff analysis method based on the identification of normal vectors on a noninferior frontier. The interactive process is implemented using a weighted minimax formulation by regulating the relative weights of objectives in a systematic manner. It is proved under a mild condition that a normal vector can be identified using the weights and Kuhn-Tucker (K-T) multipliers in the minimax formulation. Utility gradients can be estimated using local preference information such as marginal rates of substitution. The projection of a utility gradient onto a tangent plane of the noninferior frontier provides a descent direction of disutility and thereby a desirable tradeoff direction, along which tradeoff step sizes can be decided by the decision maker using an explicit tradeoff table. Necessary optimality conditions are established in terms of normal vectors and utility gradients, which can be used to guide the elicitation of local preferences and also to terminate an interactive process in a rigorous yet flexible way. This method is applicable to both linear and nonlinear (either convex or nonconvex) multiobjective optimization problems. Numerical examples are provided to illustrate the theoretical results of the paper and the implementation of the proposed interactive decision analysis process.