Directional Convexificators in Analysis

• Directional convexificators are analytic operators that modify functions to exhibit convexity in a specified direction via shear and convolution techniques.
• They are applied in geometric function theory for harmonic mappings and extended to nonsmooth optimization to derive optimality and duality conditions.
• Analytic properties such as explicit dilation bounds and chain rules support rigorous control of convexity direction in both analytic and variational frameworks.

Directional convexificators are analytic or operator constructs that, when applied to classes of harmonic or nonsmooth functions, produce new functions exhibiting convexity in a prescribed direction. They originate in geometric function theory, particularly via convolution (Hadamard product) of harmonic mappings, and extend to modern variational analysis as generalized convexity tools for nonsmooth and multiobjective optimization, providing key necessary and sufficient conditions for optimality and duality.

1. Convolution Operators as Directional Convexificators in Harmonic Mappings

Directional convexificators were first systematically analyzed in geometric function theory, particularly for harmonic mappings $f:\mathbb{D}\to\mathbb{C}$ of the unit disk. The method relies on the shear construction: given $f(z)=h(z)+\overline{g(z)}$ with analytic dilatation $\omega(z)=g'(z)/h'(z)$ and $|\omega(z)|<1$, the $\alpha$-shear $f_\alpha(z):=h(z)+e^{-2i\alpha}g(z)$ encodes the mapping's convexity direction.

The extremal analytic map

$$\phi(z) = \int_{0}^{z} \frac{d\zeta}{1-2\zeta e^{i\mu}\cos\nu+\zeta^2 e^{2i\mu}},$$

parametrized by $\mu,\nu\in\mathbb{R}$, is convex in direction $\mu$ due to its derivative's rotation and the Royster–Ziegler criterion. If $f$ and $F$ are harmonic maps with sheared-factors $f_{\mu_1}, F_{\mu_2}$ so that $f_{\mu_1}*F_{\mu_2}=\phi$, then, under local univalence and sense-preserving conditions, the convolution $f*F$ is convex in direction $-\mu$. This analysis establishes convolution operators as directional convexificators by systematically producing harmonic functions convex in specific prescribed directions via Hadamard products and directional shears (Beig et al., 2017).

2. Explicit Convolutional and Linear-Combination Constructions

Further developments provided explicit analytic models for such operators. For slanted half-plane mappings $f_\gamma$ and generalized shear operators $f_{a,\alpha}$ defined by

$$h_{a,\alpha}(z) = \frac{\tfrac{z}{1+a}-\tfrac{1}{2}e^{i\alpha}z^2}{(1-e^{i\alpha}z)^2}, \quad g_{a,\alpha}(z) = \frac{\tfrac{a e^{2i\alpha}}{1+a}z - \tfrac{1}{2}e^{3i\alpha}z^2}{(1-e^{i\alpha}z)^2},$$

the convolution $f_\gamma*f_{a,\alpha}$ is convex in direction $-(\gamma+\alpha)$ if the analytic dilatation is bounded in modulus by 1, with explicit bounds on $a$ and $n$ when $\omega(z)=e^{i\theta}z^n$. Similarly, for the right half-plane map $H$ with dilatation $\omega(z)=(a-z^2)/(1-az^2)$, $f_{a,\alpha}*H$ is convex in direction $-\alpha$ under explicit dilatation constraints. Linear combinations of a one-parameter family $f_{\alpha,n}$, each convex in the imaginary-axis direction, yield further directional convexificators through convex mixing, provided sense-preserving and local univalence are maintained (Beig et al., 2017).

In all these cases, the operator:

• modifies the domain's convexity direction by a quantifiable rotation,
• preserves univalence and sense-preservingness under explicit analytic dilatation conditions, and
• allows for analytic control of the resulting function's geometric properties.

3. Directional Convexificators in Nonsmooth Optimization

The notion of directional convexificators generalizes beyond harmonic mapping to nonsmooth and multiobjective optimization. For a lower semicontinuous function $h:\mathbb{R}^m\to\mathbb{R}\cup\{+\infty\}$ at $\bar x$, the cone of continuity directions $D(\bar x)$ comprises all $d$ such that $h(\bar x+t_k d)\to h(\bar x)$ for all $t_k\searrow0$. The directional upper convexificator (DUCF) $\partial^u_D h(\bar x)\subset\mathbb{R}^m$ is defined such that

$$h^-(\bar x; d) \le \sup_{x^*\in\partial^u_D h(\bar x)} \langle x^*, d \rangle \quad \forall d\in D,$$

with $h^-(\bar x;d)$ the lower Dini directional derivative. The DUSRCF (directional upper semi-regular convexificator) $\partial^{us}_D h(\bar x)$ replaces $h^-$ by the upper Dini derivative $h^+$. These concepts extend classical convexificators by restricting to directional cones determined by the function's continuity.

This formalism gives rise to generalized convexity notions ($\partial^u_D$–convexity, quasiconvexity, pseudoconvexity) defined via these sets, enabling precise optimality analysis even in nonsmooth, nonconvex settings (Lara et al., 22 Nov 2025).

4. Variational Analysis and Optimization: Constraint Qualification and Duality

Directional convexificators play a pivotal role in formulating generalized Abadie-type constraint qualifications in bilevel or multiobjective optimization. For a feasible set $E$ defined via equality and inequality constraints each admitting directional upper convexificators, one defines the set of active constraint convexificators and their normals. The $\partial_D$–nonsmooth Abadie-type CQ is satisfied when the polar of these convexificator sets lies in the contingent cone of $E$ restricted to the direction cone $D$.

With these notions, necessary optimality conditions are derived by representing the scalarization of the vector objective via Dini-directional derivatives and ensuring that certain linear combinations of convexificators and normals include the zero vector. Complementarity slackness and upper semicontinuity assumptions ensure robustness. Sufficient conditions leverage $\partial^u_D$–pseudoconvexity of the objectives and $\partial^u_D$–quasiconvexity of constraints, leading to global weak Pareto optimality (Lara et al., 22 Nov 2025).

The dual problem in Mond–Weir form is also formulated in terms of directional convexificators, and both weak and strong duality theorems are established under the appropriate pseudoconvexity/quasiconvexity and CQ hypotheses.

5. Analytical and Geometric Properties

A key property of directional convexificators as operators is the precise modification of the domain's convex support direction, realized through explicit analytic transformations (convolution, mixing, shearing) or, in nonsmooth analysis, through variational derivatives and polar cones of convexificator sets. The preservation or rotation of convexity direction is central: for instance, in the convolution of two harmonic maps convex in directions $\gamma_1$ and $\gamma_2$ respectively, the resulting function is convex in direction $-(\gamma_1+\gamma_2)$, assuming the required dilatation conditions. The analytic kernel used in such convolutions is typically extremal for convexity in the target direction.

In the optimization setting, the chain rule for directionally upper semi-regular convexificators enables composition and facilitates analytical tractability of nonsmooth or composite objectives and constraints. This allows extension to modern multiobjective bilevel formulations, providing a robust geometric framework underlying optimality and duality principles.

6. Illustrative Examples and Scope of Application

In harmonic mapping, explicit computation of analytic dilatation and use of convolution theorems yield concrete examples of directional convexificators acting on analytic functions or right half-plane mappings, systematically producing directionally convex outputs (Beig et al., 2017, Beig et al., 2017).

In optimization, detailed examples are constructed with piecewise or non-smooth objectives, e.g., multiobjective fractional bilevel problems. For such problems, continuity directions and directional convexificators are computed for all components at feasible solutions, and existence of multiplier tuples provides analytic verification of the necessary and sufficient conditions. Such concrete cases demonstrate the operational role of directional convexificators in both geometric function theory and variational analysis (Lara et al., 22 Nov 2025).

7. Synthesis and Outlook

Directional convexificators unify research themes in geometric function theory and modern variational analysis by providing operator-theoretic and analytic tools for the controlled modification of convexity properties in prescribed directions. In harmonic mappings, these operators enable precise control over the geometric image domain via convolution and shear techniques. In nonsmooth optimization, they underpin generalized convexity notions, constraint qualifications, and duality results, especially for nonconvex and multiobjective formulations. The analytic tractability, explicit calculations, and chain rule properties render directional convexificators central objects in contemporary analysis and geometric mapping theory. Extensions and further refinements appear in bilevel and fractional programming, highlighting ongoing research into their structural and computational properties (Beig et al., 2017, Beig et al., 2017, Lara et al., 22 Nov 2025).