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.

A relaxed projection method for variational inequalities

Abstract: This paper presents a modification of the projection methods for solving variational inequality problems. Each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself. The algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions.

Aspects of Approximation of Convex Bodies

Abstract: Publisher Summary This chapter reviews various aspects of approximation of convex bodies. Approximation of convex bodies is frequently encountered in geometric convexity, discrete geometry, the theory of finite-dimensional normed spaces, in geometric algorithms and optimization, and in the realm of engineering. Also, approximation problems in optimization arise often from more practical problems of operations research and pattern recognition. Several effective approximation algorithms formulated for convex functions or convex bodies are described in the chapter. In the former case the approximation is considered with respect to the maximum norm, in the latter case with respect to the Hausdorff metric. The chapter presents more recent developments in approximation, but many older results are also described.

A Convex Envelope Formula for Multilinear Functions

Abstract: Convex envelopes of multilinear functions on a unit hypercube are polyhedral. This well-known fact makes the convex envelope approximation very useful in the linearization of non-linear 0–1 programming problems and in global bilinear optimization. This paper presents necessary and sufficient conditions for a convex envelope to be a polyhedral function and illustrates how these conditions may be used in constructing of convex envelopes. The main result of the paper is a simple analytical formula, which defines some faces of the convex envelope of a multilinear function. This formula proves to be a generalization of the well known convex envelope formula for multilinear monomial functions.

Recurrent Neural Network for Non-Smooth Convex Optimization Problems With Application to the Identification of Genetic Regulatory Networks

Abstract: A recurrent neural network is proposed for solving the non-smooth convex optimization problem with the convex inequality and linear equality constraints. Since the objective function and inequality constraints may not be smooth, the Clarke's generalized gradients of the objective function and inequality constraints are employed to describe the dynamics of the proposed neural network. It is proved that the equilibrium point set of the proposed neural network is equivalent to the optimal solution of the original optimization problem by using the Lagrangian saddle-point theorem. Under weak conditions, the proposed neural network is proved to be stable, and the state of the neural network is convergent to one of its equilibrium points. Compared with the existing neural network models for non-smooth optimization problems, the proposed neural network can deal with a larger class of constraints and is not based on the penalty method. Finally, the proposed neural network is used to solve the identification problem of genetic regulatory networks, which can be transformed into a non-smooth convex optimization problem. The simulation results show the satisfactory identification accuracy, which demonstrates the effectiveness and efficiency of the proposed approach.