Constrained extremum problems with infinite dimensional image. Selection and saddle point

The paper deals with Image Space Analysis for constrained extremum problems having infinite dimensional image. It is shown that the introduction of selection for point- to-set maps and of quasi-multipliers allows one to establish sufficient optimality conditions for problems, where the classic ones fail.      

Constrained Extremum Problems with Infinite-Dimensional Image: Selection and Necessary Conditions

This paper deals with image space analysis for constrained extremum problems having an infinite-dimensional image. It is shown that the introduction of selection for point-to-set maps and of quasi multipliers allows one to establish optimality conditions for problems where the classical approach fails.     

Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Ω. It is known that any initial data in L²(M) can be steered to 0 in an arbitrarily small time T by applying a suitable control g in L2([0,T]×Ω), and, as T tends to 0, the norm of g grows like exp(C/T) times the norm of the data. We investigate how C depends on the geometry of Ω. We prove C?d²/4 where d is the largest distance of a point in M from Ω. When M is a segment of length L controlled at one end, we prove for some . Moreover, this bound implies where is the length of the longest generalized geodesic in M which does not intersect Ω. The control transmutation method used in proving this last result is of a broader interest.     

Experiments with primal - dual decomposition and subgradient methods for the uncapacitatied facility location problem

For optimization problems that are structured both with respect to the constraints and with respect to the variables, it is possible to use primal–dual solution approaches, based on decomposition principles. One can construct a primal subproblem, by fixing some variables, and a dual subproblem, by relaxing some constraints and king their Lagrange multipliers, so that both these problems are much easier to solve than the original problem. We study methods based on these subproblems, that do not include the difficult Benders or Dantzig-Wolfe master problems, namely primal–dual subgradient optimization methods, mean value cross decomposition, and several comtbinations of the different techniques. In this paper, these solution approaches are applied to the well-known uncapacitated facility location problem. Computational tests show that some combination methods yield near-optimal solutions quicker than the classical dual ascent method of Erlenkotter     

On Normality of Lagrange Multipliers for State Constrained Optimal Control Problems

An optimal control problem for nonlinear ODEs, subject to mixed control-state and pure state constraints is considered. Sufficient conditions are formulated, under which unique normal Lagrange multipliers exist and are given by regular functions. These conditions include pointwise linear independence of gradients of f -active constraints and controllability of the linearized state equation. Under some additional assumptions, further regularity of the multipliers is shown.     

Multipliers between Sobolev spaces and fractional differentiation

We characterize the pointwise multipliers which maps a Sobolev space to a Sobolev space in the case |s|<r<d/2.     

The form boundedness criterion for the Laplacian operator

In this paper, we characterize the class of measurable functions (or, more generally, real- or complex-valued distributions) f such that the commutator operator

Cf=[f,Δ]     

Space-time Estimates for Parabolic Type Operator and Application to Nonlinear Parabolic Equations

In this present paper we establish space-time estimates of solutions for linear parabolic type equations based on classical multipliers theory or operator semigroup theory. According to space-time estimates we first construct suitable work space L^q(0, T; L^P), moreover we study the Cauchy problem and initial boundary value problem for semilinear parabolic equation in L^q(0, T; L^P) type space.