Parameter Selection and Pre-Conditioning for a Graph Form Solver

In a recent paper, Parikh and Boyd describe a method for solving a convex optimization problem, where each iteration involves evaluating a proximal operator and projection onto a subspace. In this paper we address the critical practical issues of how to select the proximal parameter in each iteration, and how to scale the original problem variables, so as the achieve reliable practical performance. The resulting method has been implemented as an open-source software package called POGS (Proximal Graph Solver), that targets multi-core and GPU-based systems, and has been tested on a wide variety of practical problems. Numerical results show that POGS can solve very large problems (with, say, more than a billion coefficients in the data), to modest accuracy in a few tens of seconds. As just one example, a radiation treatment planning problem with around 100 million coefficients in the data can be solved in a few seconds, as compared to around one hour with an interior-point method.Comment: 28 pages, 1 figure, 1 open source implementatio

Characterizing the universal rigidity of generic frameworks

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio

On implementation of a self-dual embedding method for convex programming.

by Cheng Tak Wai, Johnny.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 59-62).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Background --- p.7Chapter 2.1 --- Self-dual embedding --- p.7Chapter 2.2 --- Conic optimization --- p.8Chapter 2.3 --- Self-dual embedded conic optimization --- p.9Chapter 2.4 --- Connection with convex programming --- p.11Chapter 2.5 --- Chapter summary --- p.15Chapter 3 --- Implementation of the algorithm --- p.17Chapter 3.1 --- The new search direction --- p.17Chapter 3.2 --- Select the step-length --- p.23Chapter 3.3 --- The multi-constraint case --- p.25Chapter 3.4 --- Chapter summary --- p.32Chapter 4 --- Numerical results on randomly generated problem --- p.34Chapter 4.1 --- Single-constraint problems --- p.35Chapter 4.2 --- Multi-constraint problems --- p.36Chapter 4.3 --- Running time and the size of the problem --- p.39Chapter 4.4 --- Chapter summary --- p.42Chapter 5 --- Geometric optimization --- p.45Chapter 5.1 --- Geometric programming --- p.45Chapter 5.1.1 --- Monomials and posynomials --- p.45Chapter 5.1.2 --- Geometric programming --- p.46Chapter 5.1.3 --- Geometric program in convex form --- p.47Chapter 5.2 --- Conic transformation --- p.48Chapter 5.3 --- Computational results of geometric optimization problem --- p.50Chapter 5.4 --- Chapter summary --- p.55Chapter 6 --- Conclusion --- p.5

LIBOR additive model calibration to swaptions markets

In the current paper, we introduce a new calibration methodology for the LIBOR market model driven by LIBOR additive processes based in an inverse problem. This problem can be splitted in the calibration of the continuous and discontinuous part, linking each part of the problem with at-the-money and in/out -of -the-money swaption volatilies. The continuous part is based on a semidefinite programming (convex) problem, with constraints in terms of variability or robustness, and the calibration of the Lévy measure is proposed to calibrate inverting the Fourier Transform

Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit

In classical inverse linear optimization, one assumes a given solution is a candidate to be optimal. Real data is imperfect and noisy, so there is no guarantee this assumption is satisfied. Inspired by regression, this paper presents a unified framework for cost function estimation in linear optimization comprising a general inverse optimization model and a corresponding goodness-of-fit metric. Although our inverse optimization model is nonconvex, we derive a closed-form solution and present the geometric intuition. Our goodness-of-fit metric, $\rho$, the coefficient of complementarity, has similar properties to $R^2$ from regression and is quasiconvex in the input data, leading to an intuitive geometric interpretation. While $\rho$ is computable in polynomial-time, we derive a lower bound that possesses the same properties, is tight for several important model variations, and is even easier to compute. We demonstrate the application of our framework for model estimation and evaluation in production planning and cancer therapy