Positivity最新文献

This paper investigates nonsmooth multiobjective fractional programming (NMFP) with inequalities and equalities constraints in real reflexive Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the Páles–Zeidan second-order directional derivative. We establish both first- and second-order strong necessary optimality conditions, which contain some new information on multipliers and imply the strong KKT necessary conditions, for a Borwein-type properly efficient solution of NMFP by utilizing generalized directional derivatives. Moreover, it derives second-order sufficient optimality conditions for NMFP under a second-order generalized convexity assumption. Additionally, we derive duality results between NMFP and its second-order dual problem under some appropriate conditions

We show that for an ideal H in an Archimedean vector lattice F the following conditions are equivalent:

• H is a projection band;

• Any collection of mutually disjoint vectors in H, which is order bounded in F, is order bounded in H;

• H is an infinite meet-distributive element of the lattice ({mathcal {I}}_{F}) of all ideals in F in the sense that (bigcap nolimits _{Jin {mathcal {J}}}left( H+ Jright) =H+ bigcap {mathcal {J}}), for any ({mathcal {J}}subset {mathcal {I}}_{F}).

Additionally, we show that if F is uniformly complete and H is a uniformly closed principal ideal, then H is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces.

This paper proposes a line search technique to solve a special class of multi-objective optimization problems in which the objective functions are supposed to be convex but need not be differentiable. This is an iterative process to determine Pareto critical points. A suitable sub-problem is proposed at every iteration of the iterative process to determine the direction vector using the sub-differential of every objective function at that point. The proposed method is verified in numerical examples. This methodology does not bear any burden of selecting suitable parameters like the scalarization methods.

In this work, we investigate inequalities of singular values and unitarily invariant norms for sums and products of matrices. First, we prove that (s^{2}big (XY^{*}big )prec _{wlog }sbig ((X^{*}X)^{q}(Y^{*}Y)(X^{*}X)^{1-q}big )), where (X, Yin M_{n}(C)) and (0<q<1). Based on this result, we present some inequalities between sum of the t-geometric mean and sum of the product of matrices. Those obtained results are the generalization of the present results. In the end, we present a singular values version of Audenaert’s inequality [1].

For a Tychonoff space X, (C^+(X)) denotes the non-negative real-valued continuous functions on X. We obtain a correlation between z-congruences on the ring C(X) and z-congruences on the semiring (C^+(X)). We give a new characterization of P-spaces via z-congruences on (C^+(X)). The z-congruences on (C^+(X)) are shown to have an algebraic nature like z-ideals. We study some topological properties of (C^+(X)) under u-topology and m-topology. It is shown that a proper ideal of (C^+(X)) is closed under m-topology if and only if it is the intersection of maximal ideals of (C^+(X)). Also, we prove that every ideal of (C^+(X)) is closed if and only if X is a P-space. We investigate the connectedness and compactness of (C^+(X)) under m-topology. It is shown that the component of (varvec{0}) is (C_psi (X)cap C^+(X)). Finally, we show that (C_m^+(X)) is locally compact, (sigma )-compact and hemicompact if and only if X is finite.

In this paper, we introduce a new F-normed space, namely Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm. Some basic properties in Orlicz–Lorentz spaces equipped with the Mazur–Orlicz F-norm are given. We find a tool to study the geometry property of Orlicz–Lorentz function spaces, the necessary and sufficient conditions for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity in Orlicz–Lorentz spaces endowed with the Mazur–Orlicz F-norm are obtained without any assumptions. The tool also can simplify the proof of the corresponding results of Orlicz–Lorentz spaces equipped with the Luxemburg norm without condition (+).

In this paper, we provide some sufficient conditions for Pettis integrability of operator valued functions that take values in ideals of compact operators on the separable Hilbert space. Additionally, we show that, in general, these conditions do not imply Bochner integrability.

We describe a natural coisometry from the Hilbert space of all Hilbert-Schmidt operators on a separable reproducing kernel Hilbert space (hbox { (RKHS)}, mathcal {H}) and onto the RKHS (mathcal {G}) associated with the squared-modulus of the reproducing kernel of (mathcal {H}). Through this coisometry, trace-class integral operators defined by general measures and the reproducing kernel of (mathcal {H}) are isometrically represented as potentials in (mathcal {G}), and the quadrature approximation of these operators is equivalent to the approximation of integral functionals on (mathcal {G}). We then discuss the extent to which the approximation of potentials in RKHSs with squared-modulus kernels can be regarded as a differentiable surrogate for the characterisation of low-rank approximation of integral operators.

In this paper, we investigate optimality conditions and duality for (varepsilon )-quasi efficient solutions of the fractional infinite multiobjective optimization problems with locally Lipschitz data. The obtained results improve or include some recent known ones. Several illustrative examples are also provided.

The starting point of this paper is the construction of a general family ( (L_{n})_{nge 1}) of positive linear operators of discrete type. Considering ((L_{n}^{k})_{kge 1}) the sequence of iterates of one of such operators, (L_{n}), our goal is to find an expression of the upper edge of the error (Vert L_{n}^{k}f-f^{*}Vert ), (fin C[0,1]), where (f^{*} ) is the fixed point of (L_{n}.) The estimate makes use of the error formula for the sequence of successive approximations in Banach’s fixed point theorem and the error of approximation of the operator (L_{n}.) Examples of special operators are inserted. Some extensions to multidimensional approximation operators are also given.