Entropy Penalized Semidefinite Programming

Low-rank methods for semidefinite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schatten-norm penalties, which are hard to implement in practice due to high computational efforts. In this paper, we propose Entropy Penalized Semi-definite programming (EP-SDP) which provides a unified framework for a wide class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit efficient numerical algorithm having (almost) linear time complexity of the gradient iteration which makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.Comment: 28th International Joint Conference on Artificial Intelligence, 201

Domain Decomposition for Stochastic Optimal Control

This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high polynomial degree to capture local phenomena. To improve the scalability of the method to problems of interest, a domain decomposition scheme is presented. By using local approximations, lower degree polynomials become sufficient, and both local and global properties of the value function are captured. The domain of the problem is split into a non-overlapping partition, with added constraints ensuring $C^1$ continuity. The Alternating Direction Method of Multipliers (ADMM) is used to optimize over each domain in parallel and ensure convergence on the boundaries of the partitions. This results in improved conditioning of the problem and allows for much larger and more complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201

Semidefinite Relaxations for Stochastic Optimal Control Policies

Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation have led to the discovery of a formulation of the value function as a linear Partial Differential Equation (PDE) for stochastic nonlinear systems with a mild constraint on their disturbances. This has yielded promising directions for research in the planning and control of nonlinear systems. This work proposes a new method obtaining approximate solutions to these linear stochastic optimal control (SOC) problems. A candidate polynomial with variable coefficients is proposed as the solution to the SOC problem. A Sum of Squares (SOS) relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in Portland, Oregon. 7 pages, colo

Exploiting Amplitude Control in Intelligent Reflecting Surface Aided Wireless Communication with Imperfect CSI

Intelligent reflecting surface (IRS) is a promising new paradigm to achieve high spectral and energy efficiency for future wireless networks by reconfiguring the wireless signal propagation via passive reflection. To reap the potential gains of IRS, channel state information (CSI) is essential, whereas channel estimation errors are inevitable in practice due to limited channel training resources. In this paper, in order to optimize the performance of IRS-aided multiuser systems with imperfect CSI, we propose to jointly design the active transmit precoding at the access point (AP) and passive reflection coefficients of IRS, each consisting of not only the conventional phase shift and also the newly exploited amplitude variation. First, the achievable rate of each user is derived assuming a practical IRS channel estimation method, which shows that the interference due to CSI errors is intricately related to the AP transmit precoders, the channel training power and the IRS reflection coefficients during both channel training and data transmission. Then, for the single-user case, by combining the benefits of the penalty method, Dinkelbach method and block successive upper-bound minimization (BSUM) method, a new penalized Dinkelbach-BSUM algorithm is proposed to optimize the IRS reflection coefficients for maximizing the achievable data transmission rate subjected to CSI errors; while for the multiuser case, a new penalty dual decomposition (PDD)-based algorithm is proposed to maximize the users' weighted sum-rate. Simulation results are presented to validate the effectiveness of our proposed algorithms as compared to benchmark schemes. In particular, useful insights are drawn to characterize the effect of IRS reflection amplitude control (with/without the conventional phase shift) on the system performance under imperfect CSI.Comment: 15 pages, 10 figures, accepted by IEEE Transactions on Communication

Multivariate GARCH estimation via a Bregman-proximal trust-region method

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the "curse of dimensionality". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimensionality of the problem. Moreover, these parameters are subjected to convoluted nonlinear constraints necessary to ensure, for instance, the existence of stationary solutions and the positive semidefinite character of the conditional covariance matrices used in the model design. So far, this problem has been addressed in the literature only in low dimensional cases with strong parsimony constraints. In this paper we propose a general formulation of the estimation problem in any dimension and develop a Bregman-proximal trust-region method for its solution. The Bregman-proximal approach allows us to handle the constraints in a very efficient and natural way by staying in the primal space and the Trust-Region mechanism stabilizes and speeds up the scheme. Preliminary computational experiments are presented and confirm the very good performances of the proposed approach.Comment: 35 pages, 5 figure

Matrix Completion via Max-Norm Constrained Optimization

Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform. In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.Comment: 33 page