Variable Smoothing for Convex Optimization Problems Using Stochastic Gradients

We aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal–dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring, and matrix factorization are provided.

On Construction of the External Frankl Nozzle Contour Using Quadratic Curvature

The aim of the article is to develop a method, an algorithm, and appropriate software for constructing the external contour of the Frankl nozzle in the supersonic part using S-shape curves. The method is based on the problem of constructing a curve with the natural parameterization. The curve passes through two given points with the given inclination angles of the tangents and provides the given inclination angle of the tangent at the point with the given abscissa [4]. To control the inflection point of the S-shaped curve, the inclination angle of the tangent at a point with the known abscissa is used. In the case, when the curvature is given by a quadratic function, the system of five nonlinear equations is formulated, among which three equations are integral. The system has five unknown variables – three coefficients of the quadratic function, the total length of the curve and the length of the curve to the point with a known abscissa. The lemma on the relation between solutions of the original and the scalable systems, in which the coordinates of the points are multiplied by the same value, is proved. Due to this lemma, it becomes possible, using the obtained solution of the well-scalable system, to find easily the corresponding solution of a bad-scalable (singular) system. To find a solution to the system, we suggest to use the modification of the r-algorithm [5] solving special problem on minimization of the nonsmooth function (the sum of the modules of the residuals of the system), under controlling of the constraints on unknown lengths, in order to guarantee their feasible values. The algorithm is implemented using the multistart method and the ralgb5a octave function [6]. It finds the best local minimum of nonsmooth function by starting the modification of the r-algorithm from a given number of starting points. The algorithm uses an analytical computation of generalized gradients of the objective function and the trapezoid rule to calculate the integrals. The computational experiment was carried out to design the fragment of supersonic part in the external contour of a Frankl-type nozzle. The efficiency of the algorithm, developed for constructing S-shape curves, is shown.

An Improved Filter Method for Nonlinear Complementarity Problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming whose objective function may be nonsmooth. For this case, we use decomposition strategy to decompose the nonsmooth function into a smooth one and a nonsmooth one. Together with filter method, we present an improved filter algorithm with decomposition strategy for solving nonlinear complementarity problem, which has better numerical results compared to the method that without the filter technique. Under mild conditions, the global convergent property is shown. In the end, the numerical example is reported.