An extension of Yuan's Lemma and its applications in optimization

We prove an extension of Yuan's Lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of [A. Baccari and A. Trad. On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. Opt., 15(2):394--408, 2005], where the classical necessary second-order optimality condition is proved under the assumption that the set of Lagrange multipliers is a bounded line segment. We prove the result under the more general assumption that the hessian of the Lagrangian evaluated at the vertices of the Lagrange multiplier set is a matrix set with at most rank 2. We apply the results to prove the classical second-order optimality condition to problems with quadratic constraints and without constant rank of the jacobian matrix

Computational Complexity of the Walrasian Equilibrium Inequalities

Recently Cherchye et al. (2011) reformulated the Walrasian equilibrium inequalities, introduced by Brown and Matzkin (1996), as an integer programming problem and proved that solving the Walrasian equilibrium inequalities is NP-hard. Following Brown and Shannon (2000), we reformulate the Walrasian equilibrium inequalities as the Hicksian equilibrium inequalities. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the Hicksian equilibrium inequalities are solvable. We show that solving the Hicksian equilibrium inequalities is equivalent to solving an NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of NP-hard minimization problems. The contribution of this paper is an approximation theorem for the NP-hard minimization, over indirect utility functions of consumers, of the maximum distance, over observations, between social endowments and aggregate Marshallian demands. In this theorem, we propose a polynomial time algorithm for computing an approximate solution to the Walrasian equilibrium inequalities, where explicit bounds on the degree of approximation are determined by observable market data

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush-Kuhn-Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine system can be written as the optimizer of a mathematical program which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with piecewise affine dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.Comment: 30 pages, 4 figure

Robust Transmission in Downlink Multiuser MISO Systems: A Rate-Splitting Approach

We consider a downlink multiuser MISO system with bounded errors in the Channel State Information at the Transmitter (CSIT). We first look at the robust design problem of achieving max-min fairness amongst users (in the worst-case sense). Contrary to the conventional approach adopted in literature, we propose a rather unorthodox design based on a Rate-Splitting (RS) strategy. Each user's message is split into two parts, a common part and a private part. All common parts are packed into one super common message encoded using a public codebook, while private parts are independently encoded. The resulting symbol streams are linearly precoded and simultaneously transmitted, and each receiver retrieves its intended message by decoding both the common stream and its corresponding private stream. For CSIT uncertainty regions that scale with SNR (e.g. by scaling the number of feedback bits), we prove that a RS-based design achieves higher max-min (symmetric) Degrees of Freedom (DoF) compared to conventional designs (NoRS). For the special case of non-scaling CSIT (e.g. fixed number of feedback bits), and contrary to NoRS, RS can achieve a non-saturating max-min rate. We propose a robust algorithm based on the cutting-set method coupled with the Weighted Minimum Mean Square Error (WMMSE) approach, and we demonstrate its performance gains over state-of-the art designs. Finally, we extend the RS strategy to address the Quality of Service (QoS) constrained power minimization problem, and we demonstrate significant gains over NoRS-based designs.Comment: Accepted for publication in IEEE Transactions on Signal Processin