Abdel Lisser

In this paper, we present a complex-valued projection neural network for solving constrained biconvex optimization problems involving real-valued functions of complex variables. The proposed framework also includes the real-valued case as a special case. We first derive the partial Karush--Kuhn--Tucker (KKT) optimality conditions for real-valued functions of complex variables and, based on these conditions, construct the corresponding neurodynamic model as an ordinary differential equation (ODE) system and establish its stability and convergence to a KKT point under suitable assumptions. Next, we consider the linear complex-valued chance-constrained optimization problem (3CP) under elliptically symmetric distributions. In the individual 3CP, we exploit distributional properties to reformulate the stochastic problem as an equivalent deterministic convex problem. In the joint chance-constrained case, we use copula theory, assuming a Gumbel--Hougaard copula, to derive a deterministic biconvex reformulation. For each case, we develop the associated neurodynamic model. Finally, we show how physics-informed neural networks (PINNs) can be used to solve the resulting ODE system. The proposed framework is further applied to minimum variance distortionless response (MVDR) beamforming, where a sector-based joint chance-constrained formulation is introduced to protect the desired response against steering-vector uncertainty and direction-of-arrival mismatch. Numerical results demonstrate the stability of the proposed approach, its convergence toward a KKT point, and its computational efficiency compared with classical optimization-based solvers.

This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimizationproblems. The probability distributions of the row vectors are not known in advance and belong to a certain distributional uncertainty set. In our paper, we study three uncertainty sets for the unknown distributions. The neurodynamic duplex is designed based on three projection equations. The main contribution of our work is to propose a neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods. We show that neural networks can be used to solve multiple instances of a problem. In the numerical experiments, we apply the proposed approach to solve a problem of shape optimisation and a telecommunication problem.

We consider a two-person zero-sum discounted stochastic game with random rewards and known transition probabilities. The players have opposite objectives and are interested in optimizing the expected discounted reward which they can obtain with a given confidence level when both the players play the worst possible move against each other. We model such a game problem by defining the chance-constrained optimization problem for each player. In this framework, risk attitudes are determined by confidence levels, where values of 0.5 or below correspond to risk-seeking behavior and values above 0.5 correspond to risk-averse behavior. When the reward vector follows a multivariate elliptically symmetric distribution, the game is equivalent to a minimax formulation. We consider the game with risk-seeking and risk-averse players separately. We show that the risk-seeking problem is equivalent to a constrained optimization of a parameterized zero-sum stochastic game and the optimal payoff of player 1 and optimal cost of player 2 can be computed using Riemann gradient sampling algorithms. Later we use the solution of the constrained optimization problem of each player to compute its optimal strategy by solving a linear programming problem. We reformulate the risk-averse problem as a discrete minimax problem. We propose an algorithm based on a linearization method and discuss its convergence properties. Alternatively, we reformulate the risk-averse problem as a second-order cone programming problem with bilinear constraints. The numerical experiments on randomly generated instances are performed to illustrate our theoretical results.

Constrained Markov decision processes (CMDPs) are used as a decision-making framework to study the long-run performance of a stochastic system. It is well known that a stationary optimal policy of a CMDP problem under discounted cost criterion can be obtained by solving a linear programming problem when running costs and transition probabilities are exactly known. In this paper, we consider a discounted cost CMDP problem where the running costs and transition probabilities are defined using random variables. Consequently, both the objective function and constraints become random. We use chance constraints to model these uncertainties and formulate the uncertain CMDP problem as a joint chance-constrained Markov decision process (JCCMDP). Under random running costs, we assume that the dependency among random constraint vectors is driven by a Gumbel–Hougaard copula. Using standard probability inequalities, we construct convex upper bound approximations of the JCCMDP problem under certain conditions on random running costs. In addition, we propose a linear programming problem whose optimal value gives a lower bound to the optimal value of the JCCMDP problem. When both running costs and transition probabilities are random, we define the latter variables as a sum of their means and random perturbations. Under mild conditions on the random perturbations and random running costs, we construct convex upper and lower bound approximations of the JCCMDP problem. We analyze the quality of the derived bounds through numerical experiments on a queueing control problem for random running costs. For the case when both running costs and transition probabilities are random, we choose randomly generated Markov decision problems called Garnets for numerical experiments.