Nonsmooth Convex Optimization Problems Research Articles

Trust region method for nonsmooth convex optimization

We propose an iterative method that solves a nonsmoothconvex optimization problem by converting the originalobjective function to a once continuously differentiablefunction by way of Moreau-Yosida regularization.The proposed method makes use of approximate functionand gradient values of the Moreau-Yosida regularizationinstead of the corresponding exact values.Under this setting, Fukushima and Qi (1996) and Raufand Fukushima (2000) proposed a proximal Newton method anda proximal BFGS method, respectively, for nonsmooth convex optimization.While these methods employ a line search strategyto achieve global convergence, the method proposed in this paperuses a trust region strategy.We establish global and superlinear convergence of the methodunder appropriate assumptions.

On Second-Order Properties of the Moreau–Yosida Regularization for Constrained Nonsmooth Convex Programs

In this paper, we attempt to investigate a class of constrained nonsmooth convex optimization problems, that is, piecewise C 2 convex objectives with smooth convex inequality constraints. By using the Moreau–Yosida regularization, we convert these problems into unconstrained smooth convex programs. Then, we investigate the second-order properties of the Moreau–Yosida regularization η. By introducing the (GAIPCQ) qualification, we show that the gradient of the regularized function η is piecewise smooth, thereby, semismooth.

Globally Convergent BFGS Method for Nonsmooth Convex Optimization1

We propose an implementable BFGS method for solving a nonsmooth convex optimization problem by converting the original objective function into a once continuously differentiable function by way of the Moreau–Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. We prove the global convergence of the proposed method under the assumption of strong convexity of the objective function.

A proximal point method for nonsmooth convex optimizationproblems in Banach spaces

In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

Numerical methods to design the reaching phase of output feedback variable structure control

Two numerical methods are developed to design a variable structure control that satisfies the reaching condition using static output feedback. The design is formulated as a nonsmooth convex optimization problem for which existing algorithms are available. It is shown how the resulting control law can be modified to be robust in the presence of parameter uncertainty or a disturbance. Numerical examples successfully demonstrate the developed techniques.

On the dual approach to the traffic assignment problem

This paper attempts to explore the possibility of solving the traffic assignment problem with elastic demands by way of its dual problem. It is shown that the dual problem can be formulated as a nonsmooth convex optimization problem of which the objective function values and subgradients are conveniently calculated by solving shortest path problems associated with the transportation network. A subgradient algorithm to solve the dual problem is presented and limited computational experience is reported. The computational results are encouraging enough to demonstrate the effectiveness of the proposed approach.

X-ray band spectra and electronic structure of carbides and nitrides of zirconium, niobium, and molibdenium

The bearing failure, generating harmful vibrations, is one of the most frequent reasons for machine breakdowns. Thus, performing bearing fault diagnosis is an essential procedure to improve the reliability of the mechanical system and reduce its operating expenses. Most of the previous studies focused on rolling bearing fault diagnosis could be categorized into two main families, kurtosis-based filter method and wavelet-based shrinkage method. Although tremendous progresses have been made, their effectiveness suffers from three potential drawbacks: firstly, fault information is often decomposed into proximal frequency bands and results in impulsive feature frequency band splitting (IFFBS) phenomenon, which significantly degrades the performance of capturing the optimal information band; secondly, noise energy spreads throughout all frequency bins and contaminates fault information in the information band, especially under the heavy noisy circumstance; thirdly, wavelet coefficients are shrunk equally to satisfy the sparsity constraints and most of the feature information energy are thus eliminated unreasonably. Therefore, exploiting two pieces of prior information (i.e., one is that the coefficient sequences of fault information in the wavelet basis is sparse, and the other is that the kurtosis of the envelope spectrum could evaluate accurately the information capacity of rolling bearing faults), a novel weighted sparse model and its corresponding framework for bearing fault diagnosis is proposed in this paper, coined KurWSD. KurWSD formulates the prior information into weighted sparse regularization terms and then obtains a nonsmooth convex optimization problem. The alternating direction method of multipliers (ADMM) is sequentially employed to solve this problem and the fault information is extracted through the estimated wavelet coefficients. Compared with state-of-the-art methods, KurWSD overcomes the three drawbacks and utilizes the advantages of both family tools. KurWSD has three main advantages: firstly, all the characteristic information scattered in proximal sub-bands is gathered through synthesizing those impulsive dominant sub-band signals and thus eliminates the dilemma of the IFFBS phenomenon. Secondly, the noises in the focused sub-bands could be alleviated efficiently through shrinking or removing the dense wavelet coefficients of Gaussian noise. Lastly, wavelet coefficients with faulty information are reliably detected and preserved through manipulating wavelet coefficients discriminatively based on the contribution to the impulsive components. Moreover, the reliability and effectiveness of the KurWSD are demonstrated through simulated and experimental signals.