A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations

Abstract

It is well known that nonlinear conjugate gradient methods are very effective for large-scale smooth optimization problems. However, their efficiency has not been widely investigated for large-scale nonsmooth problems, which are often found in practice. This paper proposes a modified Hestenes-Stiefel conjugate gradient algorithm for nonsmooth convex optimization problems. The search direction of the proposed method not only possesses the sufficient descent property but also belongs to a trust region. Under suitable conditions, the global convergence of the presented algorithm is established. The numerical results show that this method can successfully be used to solve large-scale nonsmooth problems with convex and nonconvex properties (with a maximum dimension of 60,000). Furthermore, we study the modified Hestenes-Stiefel method as a solution method for large-scale nonlinear equations and establish its global convergence. Finally, the numerical results for nonlinear equations are verified, with a maximum dimension of 100,000.

Two-Dimensional Steady-State Boundary Shape Inversion of CGM-SPSO Algorithm on Temperature Information

“…The conjugate gradient [25] transforms the inverse problem into three problems: direct question, sensitive problem, and concomitant problem. The problem is based on the assumed boundary shape (1d) and other boundary conditions, using the boundary element method to solve (1a)-(1e), then calculating the minimum value of the objective function (2).…”

“…Hong et al [21][22][23][24] optimized the PSO algorithm, which can search for more simultaneous solutions and make it possible to give an unbiased estimate that provides a better way to find the global optimal solution in search space. Yuan et al [25] introduced the improved algorithm of genetic algorithm, particle swarm optimization in two-dimensional steady-state thermal boundary problem.…”

“…The convergence of CGM and SPSO algorithm in CGM-SPSO algorithm is analyzed (see Tables 1 and 2). [25] for the detection of inner wall defects in industrial equipment is shown in Figure 1, where region is a two-dimensional space set of = 0.0, = 10.0, = 0, and = ( ). Here, both sides of = 0.0 and = 10.0 are adiabatic, and a known constant heat flow 0 is passed out by cooling at = 0 so that a constant known temperature 0 is maintained at = ( ).…”

A Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search Technique for Large-Scale Minimization Optimization Models

“…This formula is inspired by the idea of these two papers [6,8]. In recent years, lots of scholars like to study the three-term conjugate gradient formula because of its good properties [7].…”

“…The HS method has good numerical results for (1); however, the convergent theory is not interesting especially for the nonconvex function. At present, there exist many good conjugate gradients (see [6][7][8], etc.). Yuan, Wei, and Lu [9] gave a modified weak Wolfe-Powell (we called it YWL) line search for steplength designed by…”