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The paper deals with fractional optimization problems where the objective function (ratio of two functions) is defined on a Cartesian product of two real normed spaces X and Y. Within this framework, we are interested to determine the so-called partial minimizers, i.e. points in $ X \times Y $ X × Y with the property that any of its variables minimizes the objective function, restricted to this variable, with respect to the other one. While any global minimizer is obviously a partial minimizer, the reverse implication holds true only under additional assumptions (e.g. separability properties of the involved functions). By exploiting the particularities of the objective function, we deliver a Dinkelbach type algorithm for computing partial minimizers of fractional optimization problems. Further assumptions on the involved spaces and functions, such as Lipschitz-type continuity, partial Fréchet differentiability, and coercivity, enable us to establish the convergence of our algorithm to a partial minimizer. [ABSTRACT FROM AUTHOR]

The paper studies separation properties for subsets of the space N of normlinear functions on the Banach space X, i.e. the sum of a linear function and a multiple of the norm. We use as separating functionals the ones that are generated by the duality pairing [ x , p ] = p (x) = 〈 x , a 〉 + δ ‖ x ‖. We show that normlinear functions can be used to separate points from radiant or coradiant sets in X. Then we pass to separation defined by means of the pairing [ x , ⋅ ] , seen as a function defined on N . Since in this case the evaluation functionals are linear, separation describes a special subclass of convex sets in N . We characterize X-convex sets by exploiting the isomorphism between N and the space Y = X ∗ × R . We also study polarity relations defined on N . Polar sets are X-convex. And we describe what further properties are needed in order to make a subset C ⊂ Y the polar, or the reverse polar, of some set in X. For polar sets C ⊂ Y , it is possible to deduce the unique closed, radiant prepolar set U ⊂ X. To conclude, we emphasize some connections between upward (downward) X-convex sets in Y and sets of upper (resp. lower) bounds in Y. [ABSTRACT FROM AUTHOR]

Let X be a uniformly smooth and 2-uniformly convex real Banach space with dual space X ∗ . In this paper, a Halpern-type subgradient extragradient algorithm is constructed. The sequence, generated by the algorithm, converges strongly to a common solution of variational inequality and two convex minimization problems. This result is obtained as an application of a Halpern-type subgradient extragradient algorithm, for approximating a common solution of variational inequality and J-fixed points of two continuous J-pseudocontractions. The theorem proved complements, improves and unifies many recent results in the literature. Finally, numerical experiments are given to illustrate the convergence of the sequence generated by the algorithm. [ABSTRACT FROM AUTHOR]