A Property of Sobolev Spaces and Existence in Optimal Design

   Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W ₀ ^l,p (Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary value problems of arbitrary order, in arbitrary dimension and with general cost functionals.     

Orthonormality of Cardinal Chebyshev B-Spline Bases in Weighted Sobolev Spaces

Extending a recent result of Ulrich Reif on cardinal polynomial B-splines, we show that the cardinal Chebyshev B-spline basis associated with a linear differential operator with constant real coefficients is orthonormal with respect to a unique weighted Sobolev-type inner product.     

A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces

In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a p-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang (A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim. 35 (2014), 99–110), F. Scho¨pfer et al. (An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24 (2008), 055008) and many important recent results in this direction.     

Optimal Shape Design of Blazed Diffraction Gratings

The problem of designing a periodic interface between two materials in such a way that time-harmonic waves diffracted from the interface have a specified far-field pattern is studied. A minimization problem for the interface is formulated, and it is shown that solutions of constrained bounded variation exist. The differentiability of the cost functional is then studied, with no restrictions on the smoothness of the interface. Some computational issues are discussed, and finally the results of some numerical experiments are presented. Accepted 3 February 1998     

A branch and bound algorithm for bound constrained optimization problems without derivatives

In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most uninteresting parts of the search domain and concentrates on its promising subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.     

A superlinearly convergent strongly sub-feasible SSLE-type algorithm with working set for nonlinearly constrained optimization

In this paper, by means of a new efficient identification technique of active constraints and the method of strongly sub-feasible direction, we propose a new sequential system of linear equations (SSLE) algorithm for solving inequality constrained optimization problems, in which the initial point is arbitrary. At each iteration, we first yield the working set by a pivoting operation and a generalized projection; then, three or four reduced linear equations with a same coefficient are solved to obtain the search direction. After a finite number of iterations, the algorithm can produced a feasible iteration point, and it becomes the method of feasible directions. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. Under some mild conditions, the proposed algorithm is proved to be globally, strongly and superlinearly convergent. Finally, some preliminary numerical experiments are reported to show that the algorithm is practicable and effective.     

Generalized semi-infinite programming: A tutorial

This tutorial presents an introduction to generalized semi-infinite programming (GSIP) which in recent years became a vivid field of active research in mathematical programming. A GSIP problem is characterized by an infinite number of inequality constraints, and the corresponding index set depends additionally on the decision variables. There exist a wide range of applications which give rise to GSIP models; some of them are discussed in the present paper. Furthermore, geometric and topological properties of the feasible set and, in particular, the difference to the standard semi-infinite case are analyzed. By using first-order approximations of the feasible set corresponding constraint qualifications are developed. Then, necessary and sufficient first- and second-order optimality conditions are presented where directional differentiability properties of the optimal value function of the so-called lower level problem are used. Finally, an overview of numerical methods is given.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

A Spectral Conjugate Gradient Method for Unconstrained Optimization

A family of scaled conjugate gradient algorithms for large-scale unconstrained minimization is defined. The Perry, the Polak—Ribière and the Fletcher—Reeves formulae are compared using a spectral scaling derived from Raydan's spectral gradient optimization method. The best combination of formula, scaling and initial choice of step-length is compared against well known algorithms using a classical set of problems. An additional comparison involving an ill-conditioned estimation problem in Optics is presented. Accepted 22 August 2000. Online publication 26 February 2001.     

A Rigorous ODE Solver and Smale's 14th Problem

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the twenty-first century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations. July 27, 2000. Final version received: June 30, 2001.     

Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems, II

An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. We give the proof of some result stated in [AT5], and in addition we prove uniqueness of the Riccati operator, provided its final datum is suitably regular. Accepted 14 October 1998     

On the convergence of a new trust region algorithm

Summary. In this paper we present a new trust region algorithm for general nonlinear constrained optimization problems. The algorithm is based on the exact penalty function. Under very mild conditions, global convergence results for the algorithm are given. Local convergence properties are also studied. It is shown that the penalty parameter generated by the algorithm will be eventually not less than the norm of the Lagrange multipliers at the accumulation point. It is proved that the method is equivalent to the sequential quadratic programming method for all large , hence superlinearly convergent results of the SQP method can be applied. Numerical results are also reported. Received March 21, 1993     

Sobolev Exponent Estimate and Asymptotic Regularity of the M -Band Daubechies' Scaling Functions

In this paper, the direct estimate of the Sobolev exponent of refinable distributions and its application to the asymptotic estimate of the Sobolev exponent of the M -band Daubechies' scaling functions are considered. December 14, 1997. Date revised: July 1, 1998. Date accepted: July 7, 1998.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

A new family of conjugate gradient methods

In this paper we develop a new class of conjugate gradient methods for unconstrained optimization problems. A new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions. In particular, Polak–Ribiére–Polyak and Liu–Storey conjugate gradient methods are special cases of the new class of conjugate gradient methods. By estimating the local Lipschitz constant of the derivative of objective functions, we can find an adequate step size and substantially decrease the function evaluations at each iteration. Numerical results show that these new conjugate gradient methods are effective in minimizing large-scale non-convex non-quadratic functions.     

A note on the split common fixed-point problem for quasi-nonexpansive operators

Based on the very recent work by Censor and Segal (2009) [1], and inspired by Xu (2006) [9], Zhao and Yang (2005) [10], and Bauschke and Combettes (2001) [2], we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.     

A method combining norm-relaxed QP subproblems with systems of linear equations for constrained optimization

Based on the ideas of norm-relaxed sequential quadratic programming (SQP) method and the strongly sub-feasible direction method, we propose a new SQP algorithm for the solution of nonlinear inequality constrained optimization. Unlike the previous work, at each iteration, the norm-relaxed quadratic programming subproblem (NRQPS) in our algorithm only consists of the constraints corresponding to an estimate of the active set, and the high-order correction direction (used to avoid the Maratos effect) is obtained by solving a system of linear equations (SLE) which also only consists of such a subset of constraints and gradients. Moreover, the line search technique can effectively combine the initialization process with the optimization process, and therefore (if the starting point is not feasible) the iteration points always get into the feasible set after a finite number of iterations. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ), and the superlinear convergence is obtained without assuming the strict complementarity. Finally, the numerical experiments show that the proposed algorithm is effective and promising for the test problems.