On Exchange Methods for Nonlinear Semi-Infinite Programs

A new exchange method is presented for semi-infinite optimization problems with polyhedron constraints. The basic idea is to use an active set strategy as exchange rule to construct an approximate problem with finitely many constraints at each iteration. Under mild conditions, we prove that the proposed algorithm terminates in a finite number of iterations and guarantees that the solution of the resulting approximate problem at final iteration converges to the solution of the original problem within arbitrarily given tolerance. Numerical results indicate that the proposed algorithm is efficient and promising.

Fractional nonlinear stochastic heat equation with variable thermal conductivity

We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure we use the theory of Colombeau generalized uniformly continuous semigroups of operators. At the end, we study the relation of the original and the approximate problem and prove that, under certain conditions, the derivative operators appearing in these two problems are associated. Even more, we prove that under some additional conditions, solutions of the original and the approximate problem are almost certainly associated as well (assuming that the first one almost surely exists).

Sparse FIR Filter Design Based on Signomial Programming

The goal of sparse FIR filter design is to minimize the number of nonzero filter coefficients, while keeping its frequency response within specified boundaries. Such a design can be formally expressed via minimization of l0-norm of filter’s impulse response. Unfortunately, the corresponding minimization problem has combinatorial complexity. Therefore, many design methods are developed, which solve the problem approximately, or which solve the approximate problem exactly. In this paper, we propose an approach, which is based on the approximation of the l0-norm by an lp-norm with 0 < p < 1. We minimize the lp-norm using recently developed method for signomial programming (SGP). Our design starts with forming a SGP problem that describes filter specifications. The optimum solution of the problem is then found by using iterative procedure, which solves a geometric program in each iteration. The filters whose magnitude responses are constrained in minimax sense are considered. The design examples are provided illustrating that the proposed method, in most cases, results in filters with higher sparsity than those of the filters obtained by recently published methods.

Maximization of fundamental frequency and buckling load for the optimal stacking sequence design of laminated composite structures

The paper illustrates the application of a two-level approximation method to the lay-up design of laminated structures for maximization of fundamental frequency and buckling load with design constraints. Previously developed for the mass-minimization design, the approximation method was achieved by starting from an initial design of stacking sequence. Benchmark examples have verified its efficacy in dealing with the mass-reduction problems, but it does not have the capability to address the objective-maximization problems, which is mainly due to the limitation of the second-level approximate problem. In this work, this method is improved and extended for the consideration of objective maximization with more design constraints. The second-level approximate problem is reconstructed with mixed direct/reciprocal design variables, suitable for solving maximization problems. By varying different initial designs of stacking sequence and conducting repeated runs in the numerical examples, its efficiency is significantly shown after making comparisons with other methods.

A penny-shaped crack in a functionally graded layer between two half-spaces with different elastic properties

The axisymmetric static problem is considered on a pennyshaped mode I crack in an elastic inhomogeneous isotropic space. Young’s modulus of the elastic space material is non-symmetrical with respect to the crack. The procedure is proposed for approximate problem solution and determination of the stress intensity factor.

Numerical computations of the PCD method

The PCD (piecewise constant distributions) method is a discretization technique of the boundary value problems in which the unknown distribution and its derivatives are represented by piecewise constant distributions but on distinct meshes. It has the advantage of producing the most sparse stiffness matrix resulting from the approximate problem. In this contribution, we propose a general PCD triangulation by combining rectangular elements and triangular elements. We also apply this discretization technique for the elasticity problem. We end with presentation of numerical results of the proposed method for the 2D diffusion equation.