New Class Of Smoothing Functions Research Articles

Finite termination of a Newton-type algorithm based on a new class of smoothing functions for the affine variational inequality problem

In this paper, we propose a new class of smoothing functions. Some favorable properties of the functions are investigated. By using the proposed functions, the affine variational inequality problem (AVI) is reformulated as a system of parameterized smooth equations. A Newton method with a projection-type testing procedure is proposed to solve the equations. Under mild assumptions, we show that the algorithm may find a maximally complementary solution to the monotone AVI in a finite number of iterations. Preliminary numerical results indicate that the proposed smoothing functions are valuable.

A smoothing Newton method for symmetric cone complementarity problem

We first extend a new class of smoothing functions, which contains the well-known Chen-Harker-Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, for the nonlinear complementarity problem to the symmetric cone complementarity problem (SCCP). And then we present a smoothing Newton algorithm for the SCCP based on the new class of smoothing functions. Both the existence of Newton directions and the boundedness of the level set are showed for the SCCP with the Cartesian $$P_0$$ -property, which contains the monotone SCCP as a special case. The global linear convergence and locally superlinear convergence are established under a nonsingular assumption. Some numerical results for second order cone complementarity problems, a special case of SCCP, show that the proposed algorithm is effective.

A SMOOTHING NEWTON METHOD FOR NCP BASED ON A NEW CLASS OF SMOOTHING FUNCTIONS

A SMOOTHING NEWTON METHOD FOR NCP BASED ON A NEW CLASS OF SMOOTHING FUNCTIONS

An inexact smoothing-type algorithm for support vector machines

The smoothing-type algorithm has been successfully applied to solve various optimization problems. In this paper, we propose an inexact smoothing-type algorithm for solving the generalized support vector machines based on a new class of smoothing functions. In general, the smoothing-type method is designed based on some monotone line search and solving a linear system of equations exactly at each iteration. However, for the large-scale problems, solving the linear system of equations exactly can be very expensive. In order to overcome these drawbacks, solving the linear system of equations inexactly and the non-monotone line search technique are used in our smoothing-type method. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Preliminary numerical results are also reported.

A smoothing inexact Newton method for P 0 nonlinear complementarity problem

We first propose a new class of smoothing functions for the nonlinear complementarity function which contains the well-known Chen-Harker-Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P0 nonlinear complementarity problem. The global convergence and local superlinear convergence are established. Preliminary numerical results indicate the feasibility and efficiency of the algorithm.

A new class of smoothing functions and a smoothing Newton method for complementarity problems

In this paper, we introduce a new class of smoothing functions, which include some popular smoothing complementarity functions. We show that the new smoothing functions possess a system of favorite properties. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P 0 function are discussed. The Jacobian consistency of this class of smoothing functions is analyzed. Based on the new smoothing functions, we investigate a smoothing Newton algorithm for the NCP and discuss its global and local superlinear convergence. Some preliminary numerical results are reported.

Regularity theory for the fractional harmonic oscillator

In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator H σ = ( − Δ + | x | 2 ) σ , 0 < σ < 1 . More precisely, a new class of smooth functions C H k , α is defined, in which we study the action of H σ . In fact these spaces are those adapted to the operator H, hence the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite–Riesz transforms with the Hölder spaces C H k , α is needed, that we believe of independent interest.

Smoothing Newton algorithm for linear programming over symmetric cones

By using the theory of Euclidean Jordan algebras, based on a new class of smoothing functions, the Qi-Sun-Zhou’s smoothing Newton algorithm is extended to solve linear programming over symmetric cones (SCLP). The algorithm is globally convergent under suitable assumptions.