OPTIMALITY CONDITIONS AND APPROXIMATE OPTIMALITY CONDITIONS IN LOCALLY LIPSCHITZ VECTOR OPTIMIZATION

Abstract

In this paper, we study constrained locally Lipschitz vector optimization problems in which the objective and constraint spaces are Hilbert spaces, the decision space is a Banach space, the dominating cone and the constraint cone may be with empty interior. Necessary optimality conditions for this type of optimization problems are derived. A sufficient condition for the existence of approximate efficient solutions to a general vector optimization problem is presented. Necessary conditions for approximate efficient solutions to a constrained locally Lipschitz optimization problem is obtained.     

On Hölder continuity of approximate solutions to parametric equilibrium problems

We establish verifiable sufficient conditions for Hölder continuity of approximate solutions to parametric equilibrium problems, when solutions may be not unique. Many examples are provided to illustrate the need of considering approximate solutions instead of exact solutions and the essentialness of the imposed assumptions. As applications, we derive this Hölder continuity for constrained minimization, variational inequalities and fixed point problems.     

Stability of solutions for a class of nonlinear cone constrained optimization problems,part 1: Basic theory

We Gonsider a class of nonlinear cone constrained optimization problems depending on a parameter. Under the assumption of a constraint qualification, a second order sufficient optimality condition and a stability condition for the Lagrange multipliers it is shown, that for sufficiently smooth perturbations of the constraints and the objective function the optimal solutions obey a type of Lipschitz condition.     

Lipschitz continuity and local semiconcavity for exit time problems with state constraints

In the classical time optimal control problem, it is well known that the so-called Petrov condition is necessary and sufficient for the minimum time function to be locally Lipschitz continuous. In this paper, the same regularity result is obtained in the presence of nonsmooth state constraints. Moreover, for a special class of control systems we obtain a local semiconcavity result for the constrained minimum time function.     

Linear estimates of accuracy for approximate solutions of inverse problems

In this paper, we explore the question of which non-linear inverse problems, which are solved by a selected regularization method, may have so-called linear a priori accuracy estimates – that is, the accuracy of corresponding approximate solutions linearly depends on the error level of the data. In particular, we prove that if such a linear estimate exists, then the inverse problem under consideration is well posed, according to Tikhonov. For linear inverse problems, we find that the existence of linear estimates lead to, under some assumptions, the well-posedness (according to Tikhonov) on the whole space of solutions. Moreover, we consider a method for solving inverse problems with guaranteed linear estimates, called the residual method on the correctness set (RMCS). The linear a priori estimates of absolute and relative accuracy for the RMCS are presented, as well as analogous a posteriori estimates. A numerical illustration of obtaining linear a priori estimates for appropriate parametric sets of solutions using RMCS is given in comparison with Tikhonov regularization. The a posteriori estimates are calculated on these parametric sets as well.     

Uniqueness and stability in multidimensional hyperbolic inverse problems

Under a weak regularity assumption, we prove the uniqueness in multidimensional hyperbolic inverse problems with a single measurement. Moreover we show that our uniqueness results yield the best possible Lipschitz stability in L²-space in the inverse problems by means of the exact observability inequality.     

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.     

On possibility of obtaining linear accuracy evaluation of approximate solutions to inverse problems

We consider a concept of linear a priori estimate of the accuracy for approximate solutions to inverse problems with perturbed data. We establish that if the linear estimate is valid for a method of solving the inverse problem, then the inverse problem is well-posed according to Tikhonov. We also find conditions, which ensure the converse for the method of solving the inverse problem independent on the error levels of data. This method is well-known method of quasi-solutions by V. K. Ivanov. It provides for well-posed (according to Tikhonov) inverse problems the existence of linear estimates. If the error levels of data are known, a method of solving well-posed according to Tikhonov inverse problems is proposed. This method called the residual method on the correctness set (RMCS) ensures linear estimates for approximate solutions. We give an algorithm for finding linear estimates in the RMCS.     

Approximate subgradients and coderivatives in Rn

We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.Research supported by NSERC and the Shrum Endowment at Simon Fraser University.     

An Approximate Exact Penalty in Constrained Vector Optimization on Metric Spaces

In this paper, we use the penalty approach in order to study a class of constrained vector minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property iff there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems, we establish the generalized exact penalty property and obtain an estimation of the exact penalty.     

An exact penalty approach to constrained minimization problems on metric spaces

In this paper we use the penalty approach in order to study a class of constrained minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems we establish the generalized exact penalty property and obtain an estimation of the exact penalty.     

Lagrangian conditions for vector optimization in Banach spaces

We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions for weakly efficient solutions of non-convex location problems.     

Robust estimation in inverse problems via quantile coupling

In this article we consider a sequence of hierarchical space model of inverse problems.The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means.The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable,and an automatic selection principle for the nonrandom filters.This leads to the data-driven choice of weights.We also give an algorithm for its implementation.The quantile coupling inequality developed in this paper is of independent interest,because it includes the median coupling inequality in literature as a special case.     

Continuous-time method and its discretization to inverse problem of intensity-modulated radiation therapy treatment planning

We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback–Leibler divergence can be used as the Lyapunov function for the IMRT planning system.Moreover, we propose a non-negatively constrained iterative method formulated by discretizing a differential equation in the continuous method. We give proof for the convergence of a desired solution in the discretized system, theoretically. The proposed method not only reduces computational costs but also does not produce a solution with an unphysical negative radiation beam weight in solving IMRT planning inverse problems.The convergence properties of solutions for an ill-posed case are confirmed by numerical experiments using phantom data simulating a clinical setup.     

Peaceman–Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman–Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically.     

A remark on Lipschitz stability for inverse problems

An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse medium problem for the Helmholtz equation.     

Linear programming and the inverse method of images

The inverse method of images is a relatively easy way to find approximate solutions to first-passage time problems. We extend the method in several ways: We use a linear program instead of a system of linear equations; we utilize asymptotic information in addition to values at finite points; and we use a larger palette of approximating functions. These techniques enhance the scope, flexibility and accuracy of the method.     

Convergence of Newton's method and inverse function theorem in Banach space

Under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition, a proper condition which makes Newton's method converge, and an exact estimate for the radius of the ball of the inverse function theorem are given in a Banach space. Also, the relevant results on premises of Kantorovich and Smale types are improved in this paper.

     

An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation

This work studies an inverse problem of determining the first-order coefficient of degenerate parabolic equations using the measurement data specified at a fixed internal point. Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing. By the contraction mapping principle, the uniqueness of the solution for the inverse problem is proved. A numerical algorithm on the basis of the predictor-corrector method is designed to obtain the numerical solution and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown function is recovered very well. The results obtained in the paper are interesting and useful, and can be extended to other more general inverse coefficient problems of degenerate PDEs.     

Lipschitz stability estimate in the inverse Robin problem for the Stokes system

We are interested in the inverse problem of recovering a Robin coefficient defined on some non-accessible part of the boundary from available data on another part of the boundary in the non-stationary Stokes system. We prove a Lipschitz stability estimate under the a priori assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois and which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework.