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### A fixed point formulation of the $k$-means algorithm and a connection to Mumford-Shah.

Applied Mathematics E-Notes [electronic only]

### A generalization of Dubovitskij-Miliutin theorem

Acta Universitatis Carolinae. Mathematica et Physica

### A Leontief-type input-output inclusion.

Mathematical Problems in Engineering

### A new method of proof of Filippov’s theorem based on the viability theorem

Open Mathematics

Filippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation $\left|x\left(t\right)-y\left(t\right)\right|⩽r\left(t\right)=\left|{x}_{0}-y\left({t}_{0}\right)\right|{e}^{{\int }_{{t}_{0}}^{t}l\left(s\right)ds}+{\int }_{{t}_{0}}^{t}\gamma \left(s\right){e}^{{\int }_{s}^{t}l\left(\tau \right)d\tau }ds,$ where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = x ∈ ℝn: |x −y(t)| ≤ r(t), we may formulate the conclusion in Filippov’s theorem...

### A note on the relation between strong and M-stationarity for a class of mathematical programs with equilibrium constraints

Kybernetika

In this paper, we deal with strong stationarity conditions for mathematical programs with equilibrium constraints (MPEC). The main task in deriving these conditions consists in calculating the Fréchet normal cone to the graph of the solution mapping associated with the underlying generalized equation of the MPEC. We derive an inner approximation to this cone, which is exact under an additional assumption. Even if the latter fails to hold, the inner approximation can be used to check strong stationarity...

### A note on variational-type inequalities for (η,θ,δ)-pseudomonotone-type set-valued mappings in nonreflexive Banach spaces

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper the existence of solutions to variational-type inequalities problems for (η,θ,δ)- pseudomonotone-type set-valued mappings in nonreflexive Banach spaces introduced in  is considered. Presented theorem does not require a compact set-valued mapping, but requires a weaker condition 'locally bounded' for the mapping.

### A Remark on Variational Principles of Choban, Kenderov and Revalski

Bulletin of the Polish Academy of Sciences. Mathematics

We consider some variational principles in the spaces C*(X) of bounded continuous functions on metrizable spaces X, introduced by M. M. Choban, P. S. Kenderov and J. P. Revalski. In particular we give an answer (consistent with ZFC) to a question stated by these authors.

### A set oriented approach to global optimal control

ESAIM: Control, Optimisation and Calculus of Variations

We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem...

### A set oriented approach to global optimal control

ESAIM: Control, Optimisation and Calculus of Variations

We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path...

### A variant of Newton's method for generalized equations.

Revista Colombiana de Matemáticas

### A variational problem for couples of functions and multifunctions with interaction between leaves

ESAIM: Control, Optimisation and Calculus of Variations

We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy...

### Almost metric versions of Zhong's variational principle

Matematički Vesnik

### An existence theorem for set differential inclusions in a semilinear metric space

Control and Cybernetics

### An intersection theorem for set-valued mappings

Applications of Mathematics

Given a nonempty convex set $X$ in a locally convex Hausdorff topological vector space, a nonempty set $Y$ and two set-valued mappings $T:X⇉X$, $S:Y⇉X$ we prove that under suitable conditions one can find an $x\in X$ which is simultaneously a fixed point for $T$ and a common point for the family of values of $S$. Applying our intersection theorem we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.

### An Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on $\left(A,\eta \right)$-accretive framework.

Fixed Point Theory and Applications [electronic only]

### An iterative algorithm for generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces.

International Journal of Mathematics and Mathematical Sciences

### An iterative method for nonconvex equilibrium problems.

JIPAM. Journal of Inequalities in Pure &amp; Applied Mathematics [electronic only]

### An oriented coincidence index for nonlinear Fredholm inclusions with nonconvex-valued perturbations.

Abstract and Applied Analysis

### Analysis of M-stationary points to an EPEC modeling oligopolistic competition in an electricity spot market

ESAIM: Control, Optimisation and Calculus of Variations

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from the modeling of competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This first requires a structural analysis of the problem, e.g., verifying constraint qualifications. Second, the calmness property of a certain multifunction has to be verified in order to justify using M-stationarity conditions. Third,...

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