Directionally nondifferentiable metric projection

A closed convex set inR ² is constructed such that the associated metric projection onto that set is not everywhere directionally differentiable.     

Weak continuity of metric projections

A necessary and sufficient condition is found for weak continuity of a metric projection onto a finite-dimensional subspace inl _p (1<p≠2). A metric projection onto a boundedly compact set inl _p is sequentially weakly upper semicontinueus. An example is given on a convex, compact set inl ₂ onto which the metric projection is not weakly continuous. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 345–356, September, 1977.     

Remark on the Successive Projection Algorithm for the Multiple-Sets Split Feasibility Problem

Recently, assuming that the metric projection onto a closed convex set is easily calculated, Liu et al. (Numer. Func. Anal. Opt. 35:1459–1466, 2014) presented a successive projection algorithm for solving the multiple-sets split feasibility problem (MSFP). However, in some cases it is impossible or needs too much work to exactly compute the metric projection. The aim of this remark is to give a modification to the successive projection algorithm. That is, we propose a relaxed successive projection algorithm, in which the metric projections onto closed convex sets are replaced by the metric projections onto halfspaces. Clearly, the metric projection onto a halfspace may be directly calculated. So, the relaxed successive projection algorithm is easy to implement. Its theoretical convergence results are also given.     

The energy of a Kähler class on admissible manifolds

On a compact complex manifold of Kähler type, the energy E(Ω) of a Kähler class Ω is given by the squared L ²-norm of the projection onto the space of holomorphic potentials of the scalar curvature of any Kähler metric representing the said class, and any one such metric whose scalar curvature has squared L ²-norm equal to E(Ω) must be an extremal representative of Ω. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kähler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points of E(Ω) on certain admissible manifolds, producing a number of nontrivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kähler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kähler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity.     

Metric Projections and Polyhedral Spaces

We prove that if X is a Banach space and Y is a proximinal subspace of finite codimension in X such that the finite dimensional annihilator of Y is polyhedral, then the metric projection from X onto Y is lower Hausdorff semi continuous. In particular this implies that if X and Y are as above, with the unit sphere of the annihilator space of Y contained in the set of quasi-polyhedral points of X ^*, then the metric projection onto Y is Hausdorff metric continuous. Partially supported under project DST/INT/US-NSF/RPO/141/2003.     

Retraction algorithms for solving variational inequalities,pseudomonotone equilibrium problems,and fixed-point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ? _?-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.     

Lower Semicontinuity of the Solution Map to a Parametric Generalized Variational Inequality in Reflexive Banach Spaces

This paper is concerned with the study of solution stability of a parametric generalized variational inequality in reflexive Banach spaces. Under the requirements that the operator of a unperturbed problem is of class (S)_?+? and operators under consideration are pseudo-monotone and demicontinuous, we show that the solution map of a parametric generalized variational inequality is lower semicontinuous. The obtained results are proved without conditions related to the degree theory and the metric projection.     

Finding the optimal metric in classification problems with ordinal features

A method for finding the optimal distance function for the classification problem with two classes in which the objects are specified by vectors of their ordinal features is proposed. An optimal distance function is sought by the minimization of the weighted difference of the average intraclass and interclass distances. It is assumed that a specific distance function is given for each feature, which is defined on the Cartesian product of the set of integer numbers in the range from 0 to N − 1 and takes values from 0 to M. Distance functions satisfy modified metric properties. The number of admissible distance functions is calculated, which enables one to significantly reduce the complexity of the problem. To verify the appropriateness of metric optimization and to perform experiments, the nearest neighbor algorithm is used.     

Stability of a Turnpike Phenomenon for a Discrete-Time Optimal Control System

We study the structure of solutions of a discrete-time control system with a compact metric space of states X which arises in economic dynamics. This control system is described by a nonempty closed set Ω⊂X×X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous objective function v:Ω→R ¹ which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. In the present paper, we show that these turnpike properties are stable under perturbations of the objective function v.     

On the differential geometry of frame bundles

Summary The problem to ascertain an admissible structure of frame bundles is solved in this paper, presenting a tensor field H of type-(1.1) which satisfies H³ = H. It acts on the horizontal tensor field as an annihilator and on the vertical tensor field as an almost product structure. When a metric is endowed on the base manifold, it is always possible to assign the metric in the frame bundle such that its element of length obeys the Pythagorean rule when the measurement is done along horizontal and vertical distributions, and by such a general metric it can be proved that the tangent bundle of a frame bundle F(X_n) is reduced to0(n²) ×0(n); especially for n=2m, it is reduced to U(mn) ×0(n). The Lie derivative of H and the parallelism of the three lifts, horizontal, vertical and cnmplete, are examined in terms of their corresponding projection vector fields in the base space.     

A boundary parametric approximation to the linearized scalar potential magnetostatic field problem

We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary conditions can also be satisfied in the parametric framework. That is the field in the exterior of a sphere is expanded in a ‘harmonic series’ of eigenfunctions for the exterior harmonic problem. The approach is essentially a finite element method coupled with a spectral method via a boundary parametric procedure. The reformulated problem is discretized by finite element techniques which leads to a discrete parametric problem which can be solved by well conditioned iteration involving only the solution of decoupled Neumann type elliptic finite element systems and L² projection onto subspaces of spherical harmonics. Error and stability estimates given show exponential convergence in the degree of the spherical harmonics and optimal order convergence with respect to the finite element approximation for the resulting fields in L².     

J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem

This paper is a continuation of our study of the inverse monodromy problem for canonical systems of integral and differential equations which appeared in a recent issue of this journal. That paper established a parametrization of the set of all solutions to the inverse monodromy for canonical integral systems in terms of two continuous chains of matrix valued inner functions in the special case that the monodromy matrix was strongly regular (and the signature matrixJ was not definite). The correspondence between the chains and the solutions of this monodromy problem is one to one and onto. In this paper we study the solutions of this inverse problem for several different classes of chains which are specified by imposing assorted growth conditions and symmetries on the monodromy matrix and/or the matrizant (i.e., the fundamental solution) of the underlying equation. These external conditions serve to either fix or limit the class of admissible chains without computing them explicitly. This is useful because typically the chains are not easily accessible.     

Linear problem of tracking a given motion under an integral constraint on control

We consider the problem of optimally tracking a given vector function by means of a generalized projection of the trajectory of a linear controlled object with an integral constraint on the control. The deviation from a given motion is measured in the metric of the space C ^m [0, T] of continuous vector functions of appropriate dimension m. We describe a constructive method for solving this optimization problem with a given accuracy.     

On a linear problem of tracing a given motion

In the present paper, we consider the problem on the optimal tracing of a given vector function with the use of a generalized projection of the trajectory of a linear plant. The deviation of a given motion is measured in the metric C ^m [0, T] of continuous vector functions of the corresponding dimension m. We suggest an efficient method for the construction of an approximate solution of this optimization problem with given accuracy.     

On the Continuity of the Optimal Value in Parametric Linear Optimization: Stable Discretization of the Lagrangian Dual of Nonlinear Problems

This paper is focused on the stability of the optimal value, and its immediate repercussion on the stability of the optimal set, for a general parametric family of linear optimization problems in ⁿ. In our approach, the parameter ranges over an arbitrary metric space, and each parameter determines directly a set of coefficient vectors describing the linear system of constraints. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. In this way, discretization techniques for solving a nominal linear semi-infinite optimization problem may be modeled in terms of suitable parametrized problems. The stability results given in the paper are applied to the stability analysis of the Lagrangian dual associated with a parametric family of nonlinear programming problems. This dual problem is translated into a linear (semi-infinite) programming problem and, then, we prove that the lower semicontinuity of the corresponding feasible set mapping, the continuity of the optimal value function, and the upper semicontinuity of the optimal set mapping are satisfied. Then, the paper shows how these stability properties for the dual problem entail a nice behavior of parametric approximation and discretization strategies (in which an ordinary linear programming problem may be considered in each step). This approximation–discretization process is formalized by means of considering a double parameter: the original one and the finite subset of indices (grid) itself. Finally, the convex case is analyzed, showing that the referred process also allows us to approach the primal problem.Mathematics Subject Classifications (2000) Primary 90C34, 90C31; secondary 90C25, 90C05.     

Weak Chebyshev Subspaces and Continuous Selections for Parametric Projections

We examine the existence of continuous selections for the parametric projection onto weak Chebyshev subspaces. In particular, we show that if is the class of polynomial splines of degree n with the k fixed knots then the parametric projection admits a continuous selection if and only if the number of knots does not exceed the degree of splines plus one. February 15, 1996. Date revised: September 16, 1996.     

Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space

This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on the Moreau–Yosida approximation, we prove the global existence and uniqueness of the solution of the proposed evolution differential inclusion with projection and the unique solution of the proposed system is just its “slow solution” when the constrained set is defined by the affine equalities. When the outer layer objective function ψ is strongly convex, any solution of the proposed system is strongly convergent to the unique minimizer of the constrained optimization problem, while, the strongly convergence is also given when the inner layer objective function ϕ is strongly convex. Furthermore, we present some other optimization problem models, which can be solved by the proposed system. All the results obtained are new not only in the infinite dimensional Hilbert space framework but also in the finite dimensional space.     

Globally optimal solutions of max–min systems

A variety of problems in computer science, operations research, control theory, etc., can be modeled as non-linear and non-differentiable max–min systems. This paper introduces the global optimization into such systems. The criteria for the existence and uniqueness of the globally optimal solutions are established using the high matrix, optimal max-only projection set and k ^s -control vector of max–min functions. It is also shown that the global optimization can be accomplished through the partial max-only projection representation with algebraic and combinatorial features. The methods are constructive and lead to an algorithm of finding all globally optimal solutions.     

Characterization of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection

The subdual latticial cones in Hilbert spaces are characterized by the isotonicity of a generalization of the positive part mapping which can be expressed in terms of the metric projection only. Although Németh characterized the positive cone of Hilbert lattices with the metric projection and ordering only [A.B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123 (2) (2003), pp. 295–299.], this has been done for the first time for subdual latticial cones in this article. We also note that the normal generating pointed closed convex cones for which the projection onto the cone is isotone are subdual latticial cones, but there are subdual latticial cones for which the metric projection onto the cone is not isotone [G. Isac, A.B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46 (6) (1986), pp. 568–576; G. Isac, A.B. Néemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1) (1990), pp. 53–62; G. Isac, A.B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7 (4) (1990), pp. 773–802; G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153 (1) (1990), pp. 258–275; G. Isac, A.B. Németh, Isotone projection cones in Eucliden spaces, Ann. Sci. Math Québec 16 (1) (1992), pp. 35–52].