Convex optimization

About: Convex optimization is a research topic. Over the lifetime, 24906 publications have been published within this topic receiving 908795 citations. The topic is also known as: convex optimisation.

Simultaneous Variable Selection

Abstract: We propose a new method for selecting a common subset of explanatory variables where the aim is to model several response variables. The idea is a natural extension of the LASSO technique proposed by Tibshirani (1996) and is based on the (joint) residual sum of squares while constraining the parameter estimates to lie within a suitable polyhedral region. The properties of the resulting convex programming problem are analyzed for the special case of an orthonormal design. For the general case, we develop an efficient interior point algorithm. The method is illustrated on a dataset with infrared spectrometry measurements on 14 qualitatively different but correlated responses using 770 wavelengths. The aim is to select a subset of the wavelengths suitable for use as predictors for as many of the responses as possible.

Proximity control in bundle methods for convex

Abstract: Proximal bundle methods for minimizing a convex functionf generate a sequence {x k } by takingx k+1 to be the minimizer of $$\hat f^k (x) + u^k |x - x^k |^2 /2$$ , where $$\hat f^k $$ is a sufficiently accurate polyhedral approximation tof andu k > 0. The usual choice ofu k = 1 may yield very slow convergence. A technique is given for choosing {u k } adaptively that eliminates sensitivity to objective scaling. Some encouraging numerical experience is reported.

Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints

Abstract: This paper develops a new framework to analyze and design iterative optimization algorithms built on the notion of integral quadratic constraints (IQCs) from robust control theory. IQCs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. We discuss how to adapt IQC theory to study optimization algorithms, proving new inequalities about convex functions and providing a version of IQC theory adapted for use by optimization researchers. Using these inequalities, we derive numerical upper bounds on convergence rates for the Gradient method, the Heavy-ball method, Nesterov's accelerated method, and related variants by solving small, simple semidefinite programming problems. We also briefly show how these techniques can be used to search for optimization algorithms with desired performance characteristics, establishing a new methodology for algorithm design.

Toeplitz Compressed Sensing Matrices With Applications to Sparse Channel Estimation

Abstract: Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of high-dimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zero-mean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, time-invariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that time-domain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional least-squares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies.

Equilibrium programming using proximal-like algorithms

Abstract: We compute constrained equilibria satisfying an optimality condition. Important examples include convex programming, saddle problems, noncooperative games, and variational inequalities. Under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization. In the main algorithm each stage of computation requires two proximal steps, possibly using Bregman functions. One step serves to predict the next point; the other helps to correct the new prediction. To enhance practical applicability we tolerate numerical errors.