Fréchet approach in second-order optimization

We state a certain second-order sufficient optimality condition for functions defined in infinite-dimensional spaces by means of generalized Fréchet’s approach to second-order differentiability. Moreover, we show that this condition generalizes a certain second-order condition obtained in finite-dimensional spaces.     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials

Archiv der Mathematik - The works of V.A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm–Liouville problems are analytic in potentials, considered as mappings...     

Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization

In this paper, we establish new formulae for computing and/or estimating the Fréchet subdifferential of the efficient point multifunction of a parametric vector optimization problem. These formulae are presented in a broad class of conventional vector optimization problems with the presence of geometric, operator and (finite and infinite) functional constraints.     

The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems

This paper considers optimization problems with fuzzy-valued objective functions. For this class of fuzzy optimization problems we obtain Karush–Kuhn–Tucker type optimality conditions considering the concept of generalized Hukuhara differentiable and pseudo-invex fuzzy-valued functions.     

On Fréchet–Hilbert algebras

We consider Hilbert algebras with a supplementary Fréchet topology and get various extensions of the algebraic structure by using duality techniques. In particular we obtain optimal multiplier-type involutive algebras which in applications are large enough to be of significant practical use. The setting covers many situations arising from quantization rules, as those involving square-integrable families of bounded operators     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

On Fréchet’s functional equation

Fréchet’s functional equation \(\Delta _{y_1,y_2,\dots ,y_{n+1}}f=0\) plays a key role in the theory of polynomial functions. A basic theorem of Djokovi? shows that under general conditions the functional equation \(\Delta _y^{n+1}f=0\) is equivalent to Fréchet’s equation. Here we give a short alternative proof for this result using spectral synthesis.     

On the Fréchet derivative of matrix functions

In 1956 Rinehart [4] discussed the derivatives of matrix functions by considering differences ƒ(A + E) − ƒ(A) for matrices E commuting with A. In that case the derivative turned out to be ƒ′(A). In this paper the case of noncommutative A and E is treated. This leads to the Fréchet derivative of the matrix function ƒ. An explicit integral representation is obtained. Using an approach that is similar to the one in [5], a finite sum in polynomials of A is obtained. The coefficients may be computed recursively. This is useful in computing the Fréchet derivative which is needed for Newton's method to solve nonlinear matrix equations.     

On Topological Classification of Non-Archimedean Fréchet Spaces

We prove that any infinite-dimensional non-archimedean Fréchet space E is homeomorphic to where D is a discrete space with card(D) = dens(E). It follows that infinite-dimensional non-archimedean Fréchet spaces E and F are homeomorphic if and only if dens(E) = dens(F). In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field is homeomorphic to the non-archimedean Fréchet space .     

On Fredholm Operators Between Non-archimedean Fréchet Spaces

It is proved that the index of a Fredholm operator between non-Archimedean Fréchet spaces is preserved under compact perturbations. A similar result is shown for Fredholm operators between non-Archimedean polar regular LF-spaces.     

Sufficient optimality conditions in nondifferentiable optimization ∗

The concept of local cone approximations, introduced in recent years by Ester and Thierfelder, is a powerful tool for establishing optimality conditions. In this paper we show how to use it for obtaining sufficient optimality conditions in nondifferentiable optimization     

Productively Fréchet Spaces

We solve the long standing problem of characterizing the class of strongly Fréchet spaces whose product with every strongly Fréchet space is also Fréchet.     

Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization

In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz–Robinson–Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush–Kuhn–Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz–Robinson–Zowe regularity assumption.     

Inertial forward–backward methods for solving vector optimization problems

Abstract

We propose two forward–backward proximal point type algorithms with inertial/memory effects for determining weakly efficient solutions to a vector optimization problem consisting in vector-minimizing with respect to a given closed convex pointed cone the sum of a proper cone-convex vector function with a cone-convex differentiable one, both mapping from a Hilbert space to a Banach one. Inexact versions of the algorithms, more suitable for implementation, are provided as well, while as a byproduct one can also derive a forward–backward method for solving the mentioned problem. Numerical experiments with the proposed methods are carried out in the context of solving a portfolio optimization problem.     

Fréchet Means for Distributions of Persistence Diagrams

Given a distribution \(\rho \) on persistence diagrams and observations \(X_{1},\ldots ,X_{n} \mathop {\sim }\limits ^{iid} \rho \) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams \(X_{1},\ldots ,X_{n}\) . If the underlying measure \(\rho \) is a combination of Dirac masses \(\rho = \frac{1}{m} \sum _{i=1}^{m} \delta _{Z_{i}}\) then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fréchet mean computed by the algorithm given observations drawn iid from \(\rho \) . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.     

Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems

In this paper, a notion of Levitin–Polyak (LP in short) well-posedness is introduced for a vector optimization problem in terms of minimizing sequences and efficient solutions. Sufficient conditions for the LP well-posedness are studied under the assumptions of compactness of the feasible set, closedness of the set of minimal solutions and continuity of the objective function. The continuity assumption is then weakened to cone lower semicontinuity for vector-valued functions. A notion of LP minimizing sequence of sets is studied to establish another set of sufficient conditions for the LP well-posedness of the vector problem. For a quasiconvex vector optimization problem, sufficient conditions are obtained by weakening the compactness of the feasible set to a certain level-boundedness condition. This in turn leads to the equivalence of LP well-posedness and compactness of the set of efficient solutions. Some characterizations of LP well-posedness are given in terms of the upper Hausdorff convergence of the sequence of sets of approximate efficient solutions and the upper semicontinuity of an approximate efficient map by assuming the compactness of the set of efficient solutions, even when the objective function is not necessarily quasiconvex. Finally, a characterization of LP well-posedness in terms of the closedness of the approximate efficient map is provided by assuming the compactness of the feasible set.