arXiv subjects

Fernando García-Castaño

Publications and source records attributed to Fernando García-Castaño.

12 recordsLinked to original sources

arXiv2026

Norm-Cone Conjugation for Nonconvex Optimization: Scalar Duality and Exact Distance Penalization

We develop a norm-cone conjugation framework, generated by translated norm-cones \[ x\mapsto r-α\|x-x_0\|, \qquad α\geq 0. \] We show that this family generates the same abstract-convex class as the family of Lipschitz continuous concave functions, and exploit its explicit metric structure to obtain a concrete support geometry, a Fenchel--Moreau-type biconjugation theorem, and an associated norm-cone subdifferential. The main optimization consequence is a perturbation-duality theory based on partial norm-cone conjugation. Although the partial conjugate depends on a slope and a centre, the centre can be fixed at the nominal perturbation in the complete biconjugate expression without changing its value. Consequently, the effective dual problem is a scalar program whose only dual variable is the real slope $α\geq0$, even when the primal and perturbation spaces are vector-valued or infinite-dimensional. The resulting dual value is the norm-cone biconjugate of the value function at the nominal perturbation, so that strong duality is characterized by exact pointwise biconjugation. For nonconvex conic inequality problems, exact distance penalization is characterized by norm-cone supportability of the natural value function: the exact penalty parameters form its norm-cone subdifferential, and the least exact parameter is the minimal support slope. Global error bounds, equivalently global metric subregularity, provide a variational-analytic sufficient condition for exactness and hence for strong norm-cone duality. An infinite-dimensional example shows that the results remain applicable with a nonconvex objective and a nonsolid ordering cone.

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arXiv2026

Directionally Fine Families of Cones: Construction Principles, Approximation, and Separation

We introduce the notion of a directionally fine family of cones, a local geometric property expressing the possibility of selecting arbitrarily narrow cones around every direction of a normed space. We show that this local property provides a common mechanism for finite separation and approximation. In particular, directionally compact sets can be separated from closed cones by finitely many members of the family, while locally compact cones admit finite outer approximations with quantitative Hausdorff control on their sets of directions and on bounded sections. These finite constructions lead naturally to max-type positively homogeneous separators and do not require convexity of the cones to be separated. We then study two complementary realizations of the framework. For the classical family of Bishop--Phelps cones, directional fineness is characterized by the requirement that every point of the unit sphere be a denting point of the unit ball, equivalently, by property~$(G)$; in Banach spaces this amounts to rotundity together with the Kadec property. On the other hand, we introduce transversal coercivity as a general axial construction principle. It yields directionally fine families in arbitrary normed spaces, without geometric assumptions on the norm, and includes uniform axial deviation and norm-normalized axial cones as canonical models. Finally, the explicit structure of the former yields penalization and recovery results for optimization over locally compact cones, together with quantitative approximation and convergence estimates.

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arXiv2026

Separation of Regular Co-Radiant Sets via Inner Approximating Sequences, Radial Profile Functions and Applications to Optimization

We develop a new framework for the analysis of co-radiant sets based on a new notion of regularity. This framework extends the classical theory beyond co-radiant sets admitting a norm-base and provides a unified setting for separation and optimization. We introduce two new tools: inner approximating sequences and the radial profile function. The former provides a sequential approach to regularity and is the key ingredient in the proof of a general separation theorem for regular co-radiant sets, extending the existing separation theory. The latter is defined for arbitrary co-radiant sets and yields new characterizations of the existence of norm-bases, together with sequence-based characterizations of regularity under a solidness assumption on the associated cone. It also gives rise to a scalar function leading to scalarization results for approximate efficiency without additional structural assumptions on the underlying co-radiant set. Finally, we provide a variational interpretation of the separation residual by establishing radial-depth and metric error bounds. These yield sufficient conditions for approximate radial feasibility in equilibrium problems and variational inequalities, and an exact penalization principle on the zero level set of the residual.

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arXiv2026

Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization

We develop a Lagrange multiplier theory for nonconvex set-valued optimization problems under Lipschitz-type regularity conditions. Instead of classical continuous linear functionals, we introduce closed convex processes -- set-valued mappings whose graphs are closed convex cones -- as generalized Lagrange multipliers. This geometric framework extends separation principles beyond convexity and differentiability. We establish the existence of multiplier processes under verifiable assumptions, including Lipschitz regularity at a reference point, the existence of a bounded base of the ordering cone, and a nondegeneracy condition ensuring proper isolation of optimal values. These processes preserve global optimality: nondominated (respectively, minimal) solutions of the primal problem remain nondominated (respectively, minimal) in the penalized problem. In the scalar case, we obtain a one-to-one correspondence between multiplier processes and lower semicontinuous sublinear functions, yielding exact penalty formulations without additional constraint qualifications. An infinite-dimensional example shows that interiority conditions on the ordering cone, while sufficient, are not necessary. Applications to set-valued vector equilibrium problems are also discussed.

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arXiv2025

Nonlinear Separation Theorems for Co-Radiant Sets and Optimality Conditions for Approximate and Proper Approximate Solutions in Vector Optimization

This paper deals with $\varepsilon$-efficient and $\varepsilon$-properly efficient points with respect to a co-radiant set in vector optimization problems. In the first part of the paper, we establish a new nonlinear separation theorem for co-radiant sets in normed spaces. Subsequently, we obtain necessary and sufficient conditions, via scalarization, for both $\varepsilon$-efficient and $\varepsilon$-properly efficient points in a general framework, without requiring any assumptions on the co-radiant set or convexity conditions on the sets under consideration. Consequently, our results are applicable in a broader range of settings than those previously addressed in the literature.

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arXiv2025

Symmetric and Non-Symmetric Cone Separation via Bishop-Phelps Cones in Normed Spaces

In this paper, we study relationships between symmetric and non-symmetric separation of (not necessarily convex) cones by using separating cones of Bishop-Phelps type in real normed spaces. Besides extending some known results for the non-symmetric cone separation approach, we propose a new symmetric cone separation approach and establish cone separation results for it by using some cone separation results obtained for the non-symmetric cone separation approach twice (by swapping the roles of the cones). In addition to specifically emphasizing the results for the convex case, we also present some existence results for (bounded) convex bases of convex cones. Finally, we highlight some applications of symmetric and non-symmetric cone separation in optimization.

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arXiv2024

Existence and Density Theorems of Henig Global Proper Efficient Points

In this work, we provide some novel results that establish both the existence of Henig global proper efficient points and their density in the efficient set for vector optimization problems in arbitrary normed spaces. Our results do not require the assumption of convexity, and in certain cases, can be applied to unbounded sets. However, it is important to note that a weak compactness condition on the set (or on a section of it) and a separation property between the order cone and its conical neighborhoods remains necessary. The weak compactness condition ensures that certain convergence properties hold. The separation property enables the interpolation of a family of Bishop-Phelps cones between the order cone and each of its conic neighborhoods. This interpolation, combined with the proper handling of two distinct types of conic neighborhoods, plays a crucial role in the proofs of our results, which include as a particular case other results that have already been established under more restrictive conditions.

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arXiv2024

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Closed Convex Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

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arXiv2024

On dentability and cones with a large dual

In this paper, we provide some equivalences on dentability in normed spaces. Among others we prove: the origin is a denting point of a pointed cone $C$ if and only if it is a point of continuity for such a cone and $\overline{C^*-C^*}=X^*$; $x$ is a denting point of a convex set $A$ if and only if $x$ is a point of continuity and a weakly strongly extreme point of $A$. We also analize how our results help us to shed some light on several open problems in the literature.

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arXiv2024

Denting Points of Convex Sets and Weak Property ($π$) of Cones in Locally Convex Spaces

In this paper we first extend from normed spaces to locally convex spaces some characterizations of denting points in convex sets. On the other hand, we also prove that in an infrabarreled locally convex space a point in a convex set is denting if and only if it is a point of continuity and an extreme point of the closure of such a convex set under the strong topology in the second dual. The version for normed spaces of the former equivalence is new and contains, as particular cases, some known and remarkable results. We also extend from normed spaces to locally convex spaces some known characterizations of the weak property ($π$) of cones. Besides, we provide some new results regarding the angle property of cones and related. We also state that the class of cones in normed spaces having a pointed completion is the largest one for which the vertex is a denting point if and only if it is a point of continuity. Finally we analyse and answer several problems in the literature concerning geometric properties of cones which are related with density problems into vector optimization.

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arXiv2024

A Set-Valued Lagrange Theorem based on a Process for Convex Vector Programming

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

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arXiv2024

Sublinear Scalarizations for Proper and Approximate Proper Efficient Points in Nonconvex Vector Optimization

We show that under a separation property, a $\mathcal{Q}$-minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.

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