SQP algorithms for solving Toeplitz matrix approximation problem

Given an n × n matrix F, we find the nearest symmetric positive semi‐definite Toeplitz matrix T to F. The problem is formulated as a non‐linear minimization problem with positive semi‐definite Toeplitz matrix as constraints. Then a computational framework is given. An algorithm with rapid convergence is obtained by l₁ Sequential Quadratic Programming (SQP) method. Copyright © 2002 John Wiley & Sons, Ltd.     

Sparse recovery by the standard Tikhonov method

It is a common belief that the Tikhonov scheme with the -penalty fails to reconstruct a sparse structure with respect to a given system {ϕ _i }. However, in this paper we present a procedure for the sparse recovery, which is totally based on the standard Tikhonov method. This procedure consists of two steps. At first the Tikhonov scheme is used as a sieve to find the coefficients near ϕ _i , which are suspected to be non-zero. Within this step the performance of the standard Tikhonov method is controlled in some sparsity promoting space rather than in the original Hilbert one. In the second step of the proposed procedure, the coefficients with indices selected in the previous step are estimated by means of the data functional strategy. The choice of the regularization parameter is a crucial issue for both steps. We show that a recently developed parameter choice rule called the balancing principle can be effectively used here. We also present the results of computational experiments giving the evidence of the reliability of our approach.     

Parametric dual regularization for an optimal control problem with pointwise state constraints

The perturbation method is used in the dual regularization theory for a linear convex optimal control problem with a strongly convex objective functional and pointwise state constraints understood as ones in L ₂. Primary attention is given to the qualitative properties of the dual regularization method, depending on the differential properties of the value function (S-function) in the optimization problem. It is shown that the convergence of the method is closely related to the Lagrange principle and the Pontryagin maximum principle. The dual regularization scheme is shown to provide a new method for proving the maximum principle in the problem with pointwise state constraints understood in L ₂ or C. The regularized Lagrange principle in nondifferential form and the regularized Pontryagin maximum principle are discussed. Illustrative examples are presented.     

Functional Optimal Estimation Problems and Their Solution by Nonlinear Approximation Schemes

The design of state estimators for nonlinear dynamic systems affected by disturbances is addressed in a functional optimization framework. The estimator contains an innovation function that has to be chosen within a suitably defined class of functions in such a way to minimize a cost functional given by the worst-case ratio of the ℒ _p norms of the estimation error and the disturbances. Since this entails an infinite-dimensional optimization problem that under general hypotheses cannot be solved analytically, an approximate solution is sought by minimizing the cost functional over linear combinations of simple “basis functions,” represented by computational units with adjustable parameters. The selection of the parameters is made by solving a constrained nonlinear programming problem, where the constraints are given by pointwise conditions that ensure the well-definiteness of the functional and the existence of a solution. Penalty terms are introduced in the cost function to account for constraints imposed on points that result from sampling the sets to which the trajectories of the state and of the estimation error belong. To ensure an efficient covering of the sets, low-discrepancy sampling techniques are exploited that generate samples deterministically spread in a uniform way, without leaving regions of the space undersampled. Work supported by a PRIN grant from the Italian Ministry of University and Research (Project “New Techniques for the Identification and Adaptive Control of Industrial Systems”) and by the EU and the Regione Liguria trough the Regional Programs of Innovative Action of the European Regional Development Fund.     

On SIP algorithms for minimizing the mean-risk function in the multi-period single-source problem under uncertainty

We present an algorithmic framework for solving the strategic problem of assigning retailers to facilities in a multi-period single-sourcing product environment under uncertainty in the demand from the retailers and the costs of production, inventory holding, backlogging and distribution of the product. The functional to minimize is included by the expected objective function and the excess probability functional. By considering a splitting variable mathematical representation of the Deterministic Equivalent Model, we introduce several so-called Fix-and-Relax procedures that exploit the excess probability functional structure in addition to the structure of the special ordered sets related to the non-anticipativity constraints for the assignment variables. Some computational experience is reported. This research has been partially supported by the Grant TIC2003-05982-C05-05 from MCYT.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.     

Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of ₂-norms of wavelet expansion coefficients and the constraints are written in form of an ₂ automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in ₂. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns. AMS subject classification 65K10, 65N99, 93B40Angela Kunoth: This work has been supported partly by the Deutsche Forschungsgemeinschaft (SFB 611) at the Universität Bonn and by the European Communitys Human Potential Programme under contract HPRN-CT-2002-00286 Breaking Complexity.     

On the Solution Sets of Constrained Differential Inclusions with Applications

We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodory-type differential inclusion in a closed domain is an R -set provided some mild boundary conditions expressed in terms of functional constraints defining the domain are satisfied. Presented results generalize most of the existing ones. Some applications to the existence of periodic solutions as well as equilibria are given.     

The Topkis-Veinott algorithm for solving nonlinear programs with lower and upper bounded variables

In this paper we consider a nonlinear programming problem of the form to minimize f(x) subject to a x b, where f is a differentiable function on E_n and a and b are fixed vectors in E_n. We develop a variation of the feasible direction algorithm of Topkis and Veinott for solving the above problem and provide explicit expressions of the optimal directions for a family of direction-finding problems using different normalization constraints. We show that the algorithm converges to a Kuhn-Tucker point. The reported computational results indicate efficiency of the algorithm. It also indicates the strong effect of the form of the normalization constraint on convergence properties.