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.

Cutting-set methods for robust convex optimization with pessimizing oracles

Abstract: We consider a general worst-case robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worst-case analysis. With exact worst-case analysis, the method is shown to converge to a robust optimal point. With approximate worst-case analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worst-case analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warm-start techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.

Convex half-quadratic criteria and interacting auxiliary variables for image restoration

Abstract: This paper deals with convex half-quadratic criteria and associated minimization algorithms for the purpose of image restoration. It brings a number of original elements within a unified mathematical presentation based on convex duality. Firstly, the Geman and Yang (1995) and Geman and Reynolds (1992) constructions are revisited, with a view to establishing the convexity properties of the resulting half-quadratic augmented criteria, when the original nonquadratic criterion is already convex. Secondly, a family of convex Gibbsian energies that incorporate interacting auxiliary variables is revealed as a potentially fruitful extension of the Geman and Reynolds construction.

Convex recovery of a structured signal from independent random linear measurements

Abstract: This chapter develops a theoretical analysis of the convex programming method for recovering a structured signal from independent random linear measurements. This technique delivers bounds for the sampling complexity that are similar to recent results for standard Gaussian measurements, but the argument applies to a much wider class of measurement ensembles. To demonstrate the power of this approach, the chapter presents a short analysis of phase retrieval by trace-norm minimization. The key technical tool is a framework, due to Mendelson and coauthors, for bounding a nonnegative empirical process.

A Novel Recurrent Neural Network for Solving Nonlinear Optimization Problems With Inequality Constraints

Abstract: This paper presents a novel recurrent neural network for solving nonlinear optimization problems with inequality constraints. Under the condition that the Hessian matrix of the associated Lagrangian function is positive semidefinite, it is shown that the proposed neural network is stable at a Karush-Kuhn-Tucker point in the sense of Lyapunov and its output trajectory is globally convergent to a minimum solution. Compared with variety of the existing projection neural networks, including their extensions and modification, for solving such nonlinearly constrained optimization problems, it is shown that the proposed neural network can solve constrained convex optimization problems and a class of constrained nonconvex optimization problems and there is no restriction on the initial point. Simulation results show the effectiveness of the proposed neural network in solving nonlinearly constrained optimization problems.

Markov Weight Fields for face sketch synthesis

Abstract: Great progress has been made in face sketch synthesis in recent years. State-of-the-art methods commonly apply a Markov Random Fields (MRF) model to select local sketch patches from a set of training data. Such methods, however, have two major drawbacks. Firstly, the MRF model used cannot synthesize new sketch patches. Secondly, the optimization problem in solving the MRF is NP-hard. In this paper, we propose a novel Markov Weight Fields (MWF) model that is capable of synthesizing new sketch patches. We formulate our model into a convex quadratic programming (QP) problem to which the optimal solution is guaranteed. Based on the Markov property of our model, we further propose a cascade decomposition method (CDM) for solving such a large scale QP problem efficiently. Experimental results on the CUHK face sketch database and celebrity photos show that our model outperforms the common MRF model used in other state-of-the-art methods.