We investigate the problem of joint transmitter and receiver power allocation with the minimax mean square error (MSE) criterion for uplink transmissions in a multi-carrier code division multiple access (MC-CDMA) system. The objective of power allocation is to minimize the maximum MSE among all users each of which has limited transmit power. This problem is a nonlinear optimization problem. Using the Lagrange multiplier method, we derive the Karush-Kuhn-Tucker (KKT) conditions which are necessary for a power allocation to be optimal. Numerical results indicate that, compared to the minimum total MSE criterion, the minimax MSE criterion yields a higher total MSE but provides a fairer treatment across the users. The advantages of the minimax MSE criterion are more evident when we consider the bit error rate (BER) estimates. Numerical results show that the minimax MSE criterion yields a lower maximum BER and a lower average BER. We also observe that, with the minimax MSE criterion, some users do not transmit at full power. For comparison, with the minimum total MSE criterion, all users transmit at full power. In addition, we investigate robust joint transmitter and receiver power allocation where the channel state information (CSI) is not perfect. The CSI error is assumed to be unknown but bounded by a deterministic value. This problem is formulated as a semidefinite programming (SDP) problem with bilinear matrix inequality (BMI) constraints. Numerical results show that, with imperfect CSI, the minimax MSE criterion also outperforms the minimum total MSE criterion in terms of the maximum and average BERs.

Design of optimal controllers satisfying performance criteria of minimum tracking error and disturbance level for an interval system using a multi-objective evolutionary approach is proposed in this paper. Based on a worst-case design philosophy, the design problem is formulated as a minimax optimization problem, subsequently solved by a proposed two-phase multi-objective genetic algorithm (MOGA). By using two sets of interactive genetic algorithms where the first one determines the maximum (worst-case) cost function values for a given set of controller parameters while the other one minimizes the maximum cost function values passed from the first genetic algorithm, the proposed approach evolutionarily derives the optimal controllers for the interval system. To suitably assess chromosomes for their fitness in a population, root locations of the 32 generalized Kharitonov polynomials will be used to establish a constraints handling mechanism, based on which a fitness function can be constructed for effective evaluation of the chromosomes. Because of the time-consuming process that genetic algorithms generally exhibit, particularly the problem nature of minimax optimization, a parallel computation scheme for the evolutionary approach in the MATLAB-based working environment is also proposed to accelerate the design process.

The optimal design of complex infinite impulse response (IIR) two-channel quadrature mirror filter (QMF) banks with equiripple frequency response is considered. The design problem is appropriately formulated to result in a simple optimization problem. Therefore, based on a variant of Karmarkar's algorithm, we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the complex coefficients for the IIR QMFs. The effectiveness of the proposed technique is to form an appropriate Chebyshev approximation of a desired response and then find its solution from a linear subspace in several iterations. Finally, simulation results are presented for illustration and comparison.