On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the g-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.     

Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation

In this paper, we investigate the inverse problem of determining the potential of the dynamical Schrödinger equation in a bounded domain from the data of the solution in a subboundary over a time interval. Assuming that in a neighborhood of the boundary of the spatial domain, the potential is known and without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove a logarithmic stability estimate for the inverse problem with a single measurement on an arbitrarily given subboundary.     

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.     

Global Stability Results for the Weak Vector Variational Inequality

In this paper, we consider the global stability of solutions of a Weak Vector Variational Inequality in a finite-dimensional Euclidean space. Upper semi-continuity of the solution set mapping is established. And by a scalarization method, we derive a sufficient condition that guarantees the lower semi-continuity of the solution set mapping for the Weak Vector Variational Inequality     

On the stability of inner and outer approximations of a convex compact set by a ball

The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.     

Inverse problems for stationary Navier-Stokes systems

An inverse problem for a nonlinear equation in a Hilbert space is considered in which the right-hand side that is a linear combination of given functionals is found from given values of these functionals on the solution. Sufficient conditions for the existence of a solution are established, and the solution set is shown to be homeomorphic to a finite-dimensional compact set. A boundary inverse problem for the three-dimensional thermal convection equations for a viscous incompressible fluid and an inverse magnetohydrodynamics problem are considered as applications.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

Convergence Theorem for Variational Inequality in Hilbert Spaces with Applications

In this paper, we study variational inequality over the set of the common fixed points of a countable family of quasi-nonexpansive mappings. To tackle this problem, we propose an algorithm and use it to prove a strong convergence theorem under suitable conditions. As applications, we study variational inequality over the solution set of different nonlinear or linear problems, like minimization problems, split feasibility problems, convexly pseudoinverse problems, convex linear inverse problems, etc.     

Convergence analysis of an optimally accurate frozen multi-level projected steepest descent iteration for solving inverse problems

In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable.     

Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems

In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem.     

Variable-sized uncertainty and inverse problems in robust optimization

In robust optimization, the general aim is to find a solution that performs well over a set of possible parameter outcomes, the so-called uncertainty set. In this paper, we assume that the uncertainty size is not fixed, and instead aim at finding a set of robust solutions that covers all possible uncertainty set outcomes. We refer to these problems as robust optimization with variable-sized uncertainty. We discuss how to construct smallest possible sets of min–max robust solutions and give bounds on their size.A special case of this perspective is to analyze for which uncertainty sets a nominal solution ceases to be a robust solution, which amounts to an inverse robust optimization problem. We consider this problem with a min–max regret objective and present mixed-integer linear programming formulations that can be applied to construct suitable uncertainty sets.Results on both variable-sized uncertainty and inverse problems are further supported with experimental data.     

Approximation of the inverse <Emphasis Type="Italic">G</Emphasis>-frame operator

In this paper, we introduce the concept of (strong) projection method for g-frames which works for all conditional g-Riesz frames. We also derive a method for approximation of the inverse g-frame operator which is efficient for all g-frames. We show how the inverse of g-frame operator can be approximated as close as we like using finite-dimensional linear algebra.     

A generalized conditional gradient method and its connection to an iterative shrinkage method

This article combines techniques from two fields of applied mathematics: optimization theory and inverse problems. We investigate a generalized conditional gradient method and its connection to an iterative shrinkage method, which has been recently proposed for solving inverse problems. The iterative shrinkage method aims at the solution of non-quadratic minimization problems where the solution is expected to have a sparse representation in a known basis. We show that it can be interpreted as a generalized conditional gradient method. We prove the convergence of this generalized method for general class of functionals, which includes non-convex functionals. This also gives a deeper understanding of the iterative shrinkage method.     

Inverse optimization in semi-infinite linear programs

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original ones and render the given solution optimal. Ahuja and Orlin employed the absolute weighted sum metric to quantify distances between costs, and then applied duality to establish that inverse optimization reduces to another finite-dimensional LP. This paper extends this to semi-infinite linear programs (SILPs). A convergent Simplex algorithm to tackle the inverse SILP is proposed.     

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

A quasi‐boundary value regularization method for determining the heat source

In this paper, we consider the inverse problem of determining the heat source, which depends only on spatial variable in one‐dimensional heat equation in a bounded domain where data is given at some fixed time. A conditional stability result is given, and a quasi‐boundary value regularization method is also provided. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable. Copyright © 2013 John Wiley & Sons, Ltd.     

Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A^∗A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.     

On convexity of the functional for inverse problems of hyperbolic equations

In this paper, a strategy to design a functional for inverse problems of hyperbolic equations is proposed. For an inverse source problem, it is shown that the designed functional is globally strictly convex. For an inverse coefficient problem, we can only prove that it is strictly convex near true solution. This strategy can be generalized to other inverse problems, as long as Lipschitz stability is given.