This paper proposes a robust optimization model for probabilistic protection under uncertain capacity demands to minimize the total required capacity against multiple simultaneous failures of physical machines. The proposed model determines both primary and backup virtual machine allocations simultaneously under the probabilistic protection guarantee. To express the uncertainty of capacity demands, we introduce an uncertainty set that considers the upper bound of the total demand and the upper and lower bounds of each demand. The robust optimization technique is applied to the optimization model to deal with two uncertainties: failure event and capacity demand. With this technique, the model is formulated as a mixed integer linear programming (MILP) problem. To solve larger sized problems, a simulated annealing (SA) heuristic is introduced. In SA, we obtain the capacity demands by solving maximum flow problems. Numerical results show that our proposed model reduces the total required capacity compared with the conventional model by determining both primary and backup virtual machine allocations simultaneously. We also compare the results of MILP, SA, and a baseline greedy algorithm. For a larger sized problem, we obtain approximate solutions in a practical time by using SA and the greedy algorithm.

This paper presents robust optimization models for minimizing the required backup capacity while providing probabilistic protection against multiple simultaneous failures of physical machines under uncertain virtual machine capacities in a cloud provider. If random failures occur, the required capacities for virtual machines are allocated to the dedicated backup physical machines, which are determined in advance. We consider two uncertainties: failure event and virtual machine capacity. By adopting a robust optimization technique, we formulate six mixed integer linear programming problems. Numerical results show that for a small size problem, our presented models are applicable to the case that virtual machine capacities are uncertain, and by using these models, we can obtain the optimal solution of the allocation of virtual machines under the uncertainty. A simulated annealing heuristic is presented to solve large size problems. By using this heuristic, an approximate solution is obtained for a large size problem.

This paper introduces robust optimization models for minimization of the network congestion ratio that can handle the fluctuation in traffic demands between nodes. The simplest and widely used model to minimize the congestion ratio, called the pipe model, is based on precisely specified traffic demands. However, in practice, network operators are often unable to estimate exact traffic demands as they can fluctuate due to unpredictable factors. To overcome this weakness, we apply robust optimization to the problem of minimizing the network congestion ratio. First, we review existing models as robust counterparts of certain uncertainty sets. Then we consider robust optimization assuming ellipsoidal uncertainty sets, and derive a tractable optimization problem in the form of second-order cone programming (SOCP). Furthermore, we take uncertainty sets to be the intersection of ellipsoid and polyhedral sets, and considering the mirror subproblems inherent in the models, obtain tractable optimization problems, again in SOCP form. Compared to the previous model that assumes an error interval on each coordinate, our models have the advantage of being able to cope with the total amount of errors by setting a parameter that determines the volume of the ellipsoid. We perform numerical experiments to compare our SOCP models with the existing models which are formulated as linear programming problems. The results demonstrate the relevance of our models in terms of congestion ratio and computation time.

In this letter, we consider robust beamforming optimization for a multiuser multiple-input single-output system with simultaneous wireless information and power transmission (SWIPT) for the case of imperfect channel state information. Adopting the ellipsoidal uncertainty on channel vector, the robust beamforming design are reformulated as convex semi-definite programming (SDP) by rank-one relaxation. To reduce the complexity, an ellipsoidal uncertainty on channel covariance is studied to derive the equivalent form of original problem. Simulation results are provided to demonstrate the effectiveness of the proposed schemes.

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.