Optimality and duality in nonlinear programming involving semilocally B-preinvex and related functions

A nonlinear programming problem is considered where the functions involved are η-semidifferentiable. Fritz John and Karush–Kuhn–Tucker types necessary optimality conditions are obtained. Moreover, a result concerning sufficiency of optimality conditions is given. Wolfe and Mond–Weir types duality results are formulated in terms of η-semidifferentials. The duality results are given using concepts of generalized semilocally B-preinvex functions.     

Calculus of variations on time scales: weak local piecewise Crd solutions with variable endpoints

A nonlinear calculus of variations problem on time scales with variable endpoints is considered. The space of functions employed is that of piecewise rd-continuously Δ-differentiable functions (C¹_prd). For this problem, the Euler-Lagrange equation, the transversality condition, and the accessory problem are derived as necessary conditions for weak local optimality. Assuming the coercivity of the second variation, a corresponding second order sufficiency criterion is established.     

Existence and convergence theorems for generalized hybrid mappings in uniformly convex metric spaces

In this paper, we show a relationship between strictly convexity of type (I) and (II) defined by Takahashi and Talman, and we prove that any uniformly convex metric space is strictly convex of type (II). Continuity of the convex structure is also shown on a compact domain. Then, we prove the existence of a minimum point of a convex, lower semicontinuous and d-coercive function defined on a nonempty closed convex subset of a complete uniformly convex metric space. By using this property, we prove fixed point theorems for (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Using this result, we also obtain a common fixed point theorem for a countable commutative family of (α, β)-generalized hybrid mappings in uniformly convex metric spaces. Finally, we establish strong convergence of a Mann type iteration to a fixed point of (α, β)-generalized hybrid mapping in a uniformly convex metric space without assuming continuity of convex structure. Our results can be applied to obtain the existence and convergence theorems for (α, β)-generalized hybrid mappings in Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.     

Symmetries of Null Geometry in Indefinite Kenmotsu Manifolds

Null hypersurfaces have metrics with vanishing determinants and this degeneracy of these metrics leads to several difficulties. In this paper, null hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field, are studied with specific attention to locally symmetric, semi-symmetric and Ricci semi-symmetric null hypersurfaces. We show that locally symmetric and semi-symmetric null hypersurfaces are totally geodesic and parallel. These also hold for Ricci semi-symmetric null hypersurfaces, under a certain condition. We prove that, in null Einstein hypersurfaces of an indefinite Kenmotsu space form, tangent to the structure vector field, the local symmetry, semisymmetry and Ricci semi-symmetry notions are equivalent. For totally contact umbilical null hypersurfaces, we show that there are η-“Weyl” connections adapted to the induced structure on the null hypersurface.     

Nonlinear separation concerning <Emphasis Type="Italic">E</Emphasis>-optimal solution of constrained multi-objective optimization problems

This paper aims at investigating optimality conditions in terms of E-optimal solution for constrained multi-objective optimization problems in a general scheme, where E is an improvement set with respect to a nontrivial closed convex point cone with apex at the origin. In the case where E is not convex, nonlinear vector regular weak separation functions and scalar weak separation functions are introduced respectively to realize the separation between the two sets in the image space, and Lagrangian-type optimality conditions are established. These results extend and improve the convex ones in the literature.     

Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.     

Higher order duality in vector optimization over cones

In this paper higher order cone convex, pseudo convex, strongly pseudo convex, and quasiconvex functions are introduced. Higher order sufficient optimality conditions are given for a weak minimum, minimum, strong minimum and Benson proper minimum solution of a vector optimization problem. A higher order dual is associated and weak and strong duality results are established under these new generalized convexity assumptions.     

Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions

Necessary and sufficient optimality conditions are obtained for a nonlinear fractional multiple objective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions.     

(u,02)-广义次似凸映射的择一定理及其对向量优化的应用

在序线性拓扑空间中,引入（u,02）-广义次似凸映射,建立了（u,02）-广义次似凸映射的一个择一定理.最后,利用此定理,在序线性拓扑空间中得到了带广义不等式约束的向量极值问题的最优性必要条件和充分条件.     

The finite element method for a boundary value problem with strong singularity

The existence and uniqueness of the R_ν-generalized solution for the third-boundary-value problem and the non-self-adjoint second-order elliptic equation with strong singularity are established. We construct a finite element method with a basis containing singular functions. The rate of convergence of the approximate solution to the R_ν-generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.     

Nonsmooth minimax fractional programming involving η-pseudolinear functions

This article deals with the necessary and sufficient optimality conditions for a class of nonsmooth minimax fractional programming problems with locally Lipschitz η-pseudolinear functions. Utilizing these optimality criteria, we formulate two types of dual models and establish weak and strong duality results. The results of this article extend several known results from the literature to a wider class of optimization problems.     

Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications

We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush-Kuhn-Tucker conditions do not provide necessary optimality criteria. In order to obtain reasonable first order conditions under very weak assumptions, we introduce several MPVC-tailored constraint qualifications, discuss their relation, and prove an optimality condition which may be viewed as the counterpart of what is called M-stationarity in the MPEC-field.     

Strongly nonatomic densities defined by certain matrices

Drewnowski and Paúl proved about ten years ago that for any strongly nonatomic submeasure η on the power set P(?) of the set ? of all natural numbers the ideal of all null sets of η has the Nikodym property (NP). They stated the problem whether the converse is true in general. By presenting an example, Alon, Drewnowski and ?uczak proved recently that the answer is negative. Nevertheless, it is of mathematical interest to identify classes of submeasures η such that η is strongly nonatomic if and only if the set of all null sets of η has the Nikodym property. In this context, the authors proved some years ago that this equivalence holds, for instance, if one restricts the attention to the case of densities defined by regular Riesz matrices or by nonnegative regular Hausdorff methods. Also sufficient and necessary conditions in terms of the matrix coefficients are given, that the defined density is strongly nonatomic. In this paper we extend these investigations to the class of generalized Riesz matrices, introduced by Drewnowski, Florencio and Paúl in 1994.     

A note on invex problems with nonnegative variable

We provide sufficient optimality conditions involving invexity for mathematical programming problems with nonnegative variable. We also show that upon certain assumptions on the scaling function η, these problems are convex transformable. Hence certain associated convex problems should be solved. Some applications are shown.     

Minimal condition for shortest vectors in lattices of low dimension

For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG⁺, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.     

Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity

In the paper, we consider a class of nonsmooth multiobjective programming problems in which involved functions are locally Lipschitz. A new concept of invexity for locally Lipschitz vector-valued functions is introduced, called V-r-invexity. The generalized Karush–Kuhn–Tuker necessary and sufficient optimality conditions are established and duality theorems are derived for nonsmooth multiobjective programming problems involving V-r-invex functions (with respect to the same function η).     

Optimal control problems on manifolds: a dynamic programming approach

Dynamic programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimisers. In continuous time optimal control, the dynamic programming methodology has been used for problems with state space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite-dimensional C^∞ manifold. It shows that, also in a manifold setting, we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous, lower bounded, generalised solution of the Hamilton-Jacobi equation. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.     

Generalized nonsmooth invexity over cones in vector optimization

In this paper K-nonsmooth quasi-invex and (strictly or strongly) K-nonsmooth pseudo-invex functions are defined. By utilizing these new concepts, the Fritz–John type and Kuhn–Tucker type necessary optimality conditions and number of sufficient optimality conditions are established for a nonsmooth vector optimization problem wherein Clarke’s generalized gradient is used. Further a Mond Weir type dual is associated and weak and strong duality results are obtained.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

An extension of labeling techniques for finding shortest path trees

Label setting techniques are all based on Dijkstra’s condition of always scanning the node with the minimum label, which guarantees that each node will be scanned exactly once; while this condition is sufficient it is not necessary. In this paper, we discuss less restrictive conditions that allow the scanning of a node that does not have the minimum label, yet still maintaining sufficiency in scanning each node exactly once; various potential shortest path schemes are discussed, based on these conditions. Two approaches, a label setting and a flexible hybrid one are designed and implemented. The performance of the algorithms is assessed both theoretically and computationally. For comparative analysis purposes, three additional shortest path algorithms – the commonly cited in the literature – are coded and tested. The results indicate that the approaches that rely on the less restrictive optimality conditions perform substantially better for a wide range of network topologies.