A Majorization-Minimization Approach to Design of Power Transmission Networks

We propose an optimization approach to design cost-effective electrical power transmission networks. That is, we aim to select both the network structure and the line conductances (line sizes) so as to optimize the trade-off between network efficiency (low power dissipation within the transmission network) and the cost to build the network. We begin with a convex optimization method based on the paper ``Minimizing Effective Resistance of a Graph'' [Ghosh, Boyd \& Saberi]. We show that this (DC) resistive network method can be adapted to the context of AC power flow. However, that does not address the combinatorial aspect of selecting network structure. We approach this problem as selecting a subgraph within an over-complete network, posed as minimizing the (convex) network power dissipation plus a non-convex cost on line conductances that encourages sparse networks where many line conductances are set to zero. We develop a heuristic approach to solve this non-convex optimization problem using: (1) a continuation method to interpolate from the smooth, convex problem to the (non-smooth, non-convex) combinatorial problem, (2) the majorization-minimization algorithm to perform the necessary intermediate smooth but non-convex optimization steps. Ultimately, this involves solving a sequence of convex optimization problems in which we iteratively reweight a linear cost on line conductances to fit the actual non-convex cost. Several examples are presented which suggest that the overall method is a good heuristic for network design. We also consider how to obtain sparse networks that are still robust against failures of lines and/or generators.Comment: 8 pages, 3 figures. To appear in Proc. 49th IEEE Conference on Decision and Control (CDC '10

Optimization Methods for Designing Sequences with Low Autocorrelation Sidelobes

Unimodular sequences with low autocorrelations are desired in many applications, especially in the area of radar and code-division multiple access (CDMA). In this paper, we propose a new algorithm to design unimodular sequences with low integrated sidelobe level (ISL), which is a widely used measure of the goodness of a sequence's correlation property. The algorithm falls into the general framework of majorization-minimization (MM) algorithms and thus shares the monotonic property of such algorithms. In addition, the algorithm can be implemented via fast Fourier transform (FFT) operations and thus is computationally efficient. Furthermore, after some modifications the algorithm can be adapted to incorporate spectral constraints, which makes the design more flexible. Numerical experiments show that the proposed algorithms outperform existing algorithms in terms of both the quality of designed sequences and the computational complexity

Polytope of Correct (Linear Programming) Decoding and Low-Weight Pseudo-Codewords

We analyze Linear Programming (LP) decoding of graphical binary codes operating over soft-output, symmetric and log-concave channels. We show that the error-surface, separating domain of the correct decoding from domain of the erroneous decoding, is a polytope. We formulate the problem of finding the lowest-weight pseudo-codeword as a non-convex optimization (maximization of a convex function) over a polytope, with the cost function defined by the channel and the polytope defined by the structure of the code. This formulation suggests new provably convergent heuristics for finding the lowest weight pseudo-codewords improving in quality upon previously discussed. The algorithm performance is tested on the example of the Tanner [155, 64, 20] code over the Additive White Gaussian Noise (AWGN) channel.Comment: 6 pages, 2 figures, accepted for IEEE ISIT 201