Solution Of Minimization Problem Research Articles

Iterated Hard Shrinkage for Minimization Problems with Sparsity Constraints

A new iterative algorithm for the solution of minimization problems in infinite-dimensional Hilbert spaces which involve sparsity constraints in form of $\ell^{p}$-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise, and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method. Convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed. This leads to strong convergence of the iterates with convergence rates $\mathcal{O}(n^{-1/2})$ and $\mathcal{O}(\lambda^n)$ for $p=1$ and $1 < p \leq 2$, respectively. Numerical experiments on image deblurring, backwards heat conduction, and inverse integration are given.

The Lagrange method for the regularization of discrete ill-posed problems

In many science and engineering applications, the discretization of linear ill-posed problems gives rise to large ill-conditioned linear systems with the right-hand side degraded by noise. The solution of such linear systems requires the solution of minimization problems with one quadratic constraint, depending on an estimate of the variance of the noise. This strategy is known as regularization. In this work, we propose a modification of the Lagrange method for the solution of the noise constrained regularization problem. We present the numerical results of test problems, image restoration and medical imaging denoising. Our results indicate that the proposed Lagrange method is effective and efficient in computing good regularized solutions of ill-conditioned linear systems and in computing the corresponding Lagrange multipliers. Moreover, our numerical experiments show that the Lagrange method is computationally convenient. Therefore, the Lagrange method is a promising approach for dealing with ill-posed problems.

An improved finite point method for tridimensional potential flows

At the local level, successful meshless techniques such as the Finite Point Method must have two main characteristics: a suitable geometrical support and a robust numerical approximation built on the former. In this article we develop the second condition and present an alternative procedure to obtain shape functions and their derivatives from a given cloud of points regardless of its geometrical features. This procedure, based on a QR factorization and an iterative adjust of local approximation parameters, allows obtaining a satisfactory minimization problem solution, even in the most difficult cases where usual approaches fail. It is known that high-order meshless constructions need to include a large number of points in the local support zone and this fact turns the approximation more dependent on the size, shape and spatial distribution of the local cloud of points. The proposed procedure also facilitates the construction of high-order approximations on generic geometries reducing their dependence on the geometrical support where they are based. Apart from the alternative solution to the minimization problem, the behaviour of high-order Finite Point approximations and the overall performance of the proposed methodology are shown by means of several numerical tests.

Homogenization of Gradient-Constrained Problems with Respect to Periodic Measures

We study the asymptotic behavior of solutions of minimization problems of integral functionals with integrands satisfying p-growth conditions, with respect to constraints that are imposed on gradients of admissible functions on a periodic disperse set and with respect to periodic measures that are assumed to be nondegenerate and satisfying the Poincare inequality.

A regularized continuous projection minimization method of the second order with a variable metric

A regularized method for solution of minimization problems with imperfect output data in a Hilbert space is proposed and investigated. The method is based on a continuous second-order projection method with a variable metric. The sufficient convergence conditions are given and the stopping rule of the method is constructed.

Well-Posedness of Two Nonrigid Multimodal Image Registration Methods

Image registration methods may be designed as solutions of minimization problems on a set of geometric deformations. In the nonrigid case, solving these problems often means computing the steady state of a system of evolution equations involving the gradient of the error criterion, defined from the Euler--Lagrange equations of the corresponding minimization problem. The well-posedness of the registration methodrequires showing the existence of a solution to the minimization problem, as well as that of a stable solution of the evolution equations derived from it. We provide such proofs in the case where the error criterion is derived from two different statistical similarity measures: global mutual information and local cross-covariance. We also describe our numerical implementation for solving the corresponding evolution equations and show examples of registrations of real 2D and 3D images achieved with these algorithms. The proofs are quite general and can be applied to most of the known nonrigid image...

Numerical solution of the inverse problem of determining an unknown source term in a two-dimensional heat equation

A numerical procedure for an inverse problem of determination of unknown source term in two-dimensional parabolic equation is presented. The approach of the proposed method is to approximate unknown function by a piecewise linear function whose coefficients are determined from the solution of minimisation problem based on the overspecified data. Some numerical examples are presented.

Numerical solution of the inverse problem of determining an unknown source term in a heat equation

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. A numerical examples are presented.

Numerical procedure for the determination of an unknown coefficients in parabolic equations

A numerical procedure for an inverse problem of determination of unknown coefficients in a class of parabolic differential equations is presented. The approach of the proposed method is to approximate unknown coefficients by a piecewise linear function whose coefficients are determined from the solution of minimization problem based on the overspecified data. Some numerical examples are presented.

Numerical method of identification of an unknown source term in aheat equation

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

A Truncated Newton Algorithm for Large Scale Box Constrained Optimization

A method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the approximate solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity assumption. The algorithm proposed is based on a simple, smooth unconstrained reformulation of the bound constrained problem and may produce a sequence of points that are not feasible. Numerical results and comparison with existing codes are reported.

Existence of solutions of minimization problems with an increasing cost function and porosity

We consider the minimization problem f(x) → min, x ∈ K, where K is a closed subset of an ordered Banach space X and f belongs to a space of increasing lower semicontinuous functions on K. In our previous work, we showed that the complement of the set of all functions f, for which the corresponding minimization problem has a solution, is of the first category. In the present paper we show that this complement is also a σ‐porous set.

Viscosity Solutions of Minimization Problems

Viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis. Variational convergences for sequences of functions (epi-convergence, $\Gamma $-convergence, Mosco-convergence) and for sequences of operators (graph-convergence) provide a flexible tool for such questions. It is proved, in a rather large setting, that the solutions of the approximate problems converge to a ``viscosity solution'' of the original problem, that is, a solution that is minimal among all the solutions with respect to some viscosity criteria. Various examples coming from mathematical programming, calculus of variations, semicoercive elliptic equations, phase transition theory, Hamilton--Jacobi equations, singular perturbations, and optimal control theory are considered.

Coupling the auxiliary problem principle with descent methods of pseudoconvex programming

The descent auxiliary problem method allows one to find the solution of minimization problems by solving a sequence of auxiliary problems which incorporate a linesearch strategy. We derive the basic algorithm and study its convergence properties within the framework of infinite dimensional pseudoconvex minimization. We also introduce a partial descent type auxiliary problem method which partially linearizes the objective functional and includes the auxiliary term only for of subset of variables. Numerous examples of this very general scheme are provided.

Iteratively reweighted least squares: A comparison of several single step algorithms for linear models

Various algorithms for the computation of linear M-estimates for convex as well as for nonconvex criterion functions are considered. In the convex case a scaling parameters is estimated together with the parameter vectorx as a solution of some nonlinear minimization problem. Several single step algorithms for the solution of this minimization problem are investigated. On the base of known procedures new methods are developed, which use locally majorizing approximations of the objective function, “exact” line search and efficient “updating” techniques. These algorithms also turn out to be efficient for the computation of M-estimates which are defined by nonconvex criterion functions and a constant scaling parameter. The numerical efficiency of the different procedures is compared by means of 30 simulated test problems.

Finite dimensional approximation to solutions of minimization problems in functional spaces

In this paper we consider minimization problems whose objectives are defined on functional spaces. The integral global optimization technique is applied to characterize a global minimum as the limit of a sequence of approximating solutions on finite dimensional subspaces. Necessary and sufficient optimality conditions are presented. A variable measure algorithm is proposed to find such approximating solutions. Examples are presented to illustrate the variable measure method

A Class of penalty functions for optimization problema with bound constraints

In this paper we propose a new class of continuously differentiable globally exact penalty functions for the solution of minimization problems with simple bounds on some (all) of the variables. The penalty functions in this class fully exploit the structure of the problem and are easily computable. Furthermore we introduce a simple updating rule for the penalty parameter that can be used in conjunction with unconstrained minimization techniques to solve the original problem.

On stability of approximate solutions of minimization problems

In this paper, we investigate stability of the optimal value function and the set of approximate solutions of constrained minimization problems under various perturbations of the objective function and the constraint set. Some interesting applications to approximation theory are also explored.

C-programming problems: a class of non-linear optimization problems

A method is introduced for the solution of minimization problems possessing objective functions of the form g( x)+ ϕ( u( x)), where ϕ is differentiable and concave. It is shown that a solution to such problems can be obtained through the solution of the Lagrangian problem whose objective function has the form g( x)+ λu( x).

The concentration-compactness principle in the Calculus of Variations. The Locally compact case, part 2

In this paper (sequel of Part 1) we investigate further applications of the concentration-compactness principle to the solution of various minimization problems in unbounded domains. In particular we present here the solution of minimization problems associated with nonlinear field equations.