Convex & Variational Analysis

• Convex and variational analysis is a framework that studies convex sets, functions, and duality principles to address optimization problems.
• It employs tools such as Fenchel conjugates, subdifferentials, and variational convexity to analyze stability and sensitivity in complex models.
• The methodologies underpin numerical schemes and algorithms applicable in engineering, economics, machine learning, and dynamical systems.

Convex and variational analysis is a foundational domain at the intersection of functional analysis, optimization, geometric measure theory, and nonsmooth calculus. It provides the theoretical backbone for understanding and solving a broad array of problems in mathematics, engineering, economics, statistics, and applied sciences. Modern convex and variational analysis unifies the classical calculus of convex functions and sets with a robust toolkit for nonconvex, nonsmooth, infinite-dimensional, and composite models, facilitating rigorous structural analysis, duality theories, algorithmic development, and stability assessments.

1. Foundational Principles and Objects

Convex analysis studies convex sets and convex functions, leveraging their geometric and analytic properties to derive powerful existence, stability, and duality results. A set $C$ in a vector space is convex if its epigraph is convex; a function $f$ is convex if its epigraph

$$\operatorname{epi} f := \{(x, \alpha) \mid \alpha \ge f(x)\}$$

is convex. The convex conjugate (Fenchel dual) of $f:X\to\mathbb{R}\cup\{+\infty\}$ is $$f^*(x^*) = \sup_{x\in X}\{\langle x^*,x\rangle - f(x)\}$$.

Key geometric objects include:

• Support functions $$\sigma_C(x^*) = \sup_{x \in C} \langle x^*, x\rangle$$, always convex and weak*-lower semicontinuous.
• Tangent and normal cones: Given $C$ and $\bar x \in C$, $T_C(\bar x)$ is the Bouligand tangent cone; $N_C(\bar x)$ is its polar in the dual space.

Subdifferentials generalize derivatives to convex and nonconvex settings. For proper $f$, the (convex) subdifferential at $x$ is

$$\partial f(x) = \{ x^* \in X^* \mid f(u) \ge f(x) + \langle x^*, u-x\rangle,\ \forall u \in X \}.$$

Classical calculus extends via geometric arguments—such as the convex extremal principle and intersection rules—to sum, chain, infimal convolution, and maximum operations (Mordukhovich et al., 2016).

Generalized differentiation is further refined for nonsmooth functions via the (Mordukhovich) limiting subdifferential and its second-order (coderivative) analogs, pivotal for stability and sensitivity analysis (Mordukhovich et al., 2015, Khanh et al., 2022).

2. Variational Convexity and Generalized Convexity Notions

Rockafellar’s notion of variational convexity captures local convex-like properties essential for stability and computation, generalizing both classical convexity and strong convexity to local models. A function $f$ is variationally convex at $\bar{x}$ for $\bar{v}\in\partial f(\bar{x})$ if there exists a convex neighborhood where $f$ and a convex surrogate $\varphi$ agree on subdifferential graphs and function values up to a level set (Khanh et al., 2022, Khanh et al., 2022, Khanh et al., 2023).

Formally, $f$ is variationally convex at $\bar x$ for $\bar x^*$ if there is a convex lsc $\varphi$ such that

$$(U_\varepsilon\times V)\cap \operatorname{gph}\partial f = (U_\varepsilon\times V)\cap \operatorname{gph}\partial\varphi, \quad f(\bar{x}) = \varphi(\bar{x}).$$

The strong variant replaces $\varphi$ with a strongly convex model.

Major results:

• Variational (strong) convexity is characterized by local monotonicity (or strong monotonicity) of the subdifferential mapping or, equivalently, by convexity (or strong convexity) of localized Moreau envelopes (Khanh et al., 2022, Khanh et al., 2022, Khanh et al., 2023).
• In Banach spaces, variational convexity is linked to maximal monotonicity and local convexity of Moreau envelopes under prox-regularity (Khanh et al., 2022, Khanh et al., 2023).

Applications include variational sufficiency in optimization, tilt stability of minimizers, and second-order condition formulations (Khanh et al., 2022, Gfrerer, 25 Aug 2024).

3. Calculus, Duality, and Variational Principles

Convex and variational analysis underpins a general duality paradigm via Fenchel conjugates, separation theorems, and abstract variational principles. The calculus rules include:

• Infimal convolution: $(f\Box g)(x) = \inf_{u+v=x}\{f(u)+g(v)\}$.
• Sum rules: under mild qualification, $\partial(f+g)(x) = \partial f(x) + \partial g(x)$, $(f+g)^* = f^*\Box g^*$.
• Chain rules: $$(g\circ A)^*(x^*) = \inf\{g^*(y^*) \mid A^*y^* = x^*\}$$, $\partial(g\circ A)(x) = A^*\partial g(Ax)$.
• Optimal-value function: $$\partial p(x) = \bigcup_{y\in M(x)} \{x^*+D^*F(x,y)(y^*) \mid (x^*,y^*)\in\partial\varphi(x,y)\}$$ where $p(x)=\inf_{y\in F(x)}\varphi(x,y)$.

Thermodynamic formalism and entropy–pressure dualities are constructed by convex-variational approaches on spaces of finitely additive set functions, with all equilibrium states captured as subgradients of pressure functions (Bis et al., 2020).

Strong duality frameworks for nonconvex variational problems can be obtained by convexification—DC decompositions, relaxation, or infinite-dimensional convex programming—leading to dual SDPs and equivalence of measure- and multiplier-based relaxation methods (Botelho, 2021, Fantuzzi, 2019).

4. Convexity-like Constraints and Numerical Variational Schemes

Directional and polyhedral approximations to non-polyhedral convex constraint cones enable the tractable numerical solution of variational problems with convexity constraints (Oberman, 2011, Mérigot et al., 2014). Key mechanisms:

• Polyhedral approximations enforce finite directional convexity constraints via second differences along discrete directions.
• Under grid refinement and sufficient directionality, the discrete constraint sets approximate the true convexity cone in the Hausdorff sense. Error and convergence analysis is explicit.
• Proximal splitting algorithms (e.g., SDMM) can decompose large-scale discretized problems into block subproblems, supporting efficient, parallelizable algorithms applicable to 2D/3D grids (Mérigot et al., 2014).
• Example domains include $L^2$ projection onto convex functions, principal-agent problems with c-convexity, and optimization over support functions of convex bodies.

Spectral-variational methods extend this by developing a unified calculus for convex (and more generally spectral) functions defined by their matrix eigenvalues or singular values, characterizing Fréchet, limiting, and Clarke subdifferentials, and providing new Lidskiĭ-type perturbation theorems for spectral decomposition systems (Bùi et al., 13 Oct 2025).

5. Second-Order Analysis and Piecewise Linear Functions

Second-order subdifferential and coderivative calculus for convex extended-real- and piecewise-linear functions enables explicit, algebraic criteria for stability, strong optimality, and sensitivity (Mordukhovich et al., 2015).

• The (Mordukhovich) second-order subdifferential $\partial^2 f(x\mid v)(u)$ obtains a full polyhedral (or SC-basis) representation for convex piecewise linear functions, allowing concrete verification of second-order variational sufficiency.
• Exact primary second-order sum rules hold in this setting: for $p_1,p_2$ convex piecewise-linear,

$$\partial^2(p_1+p_2)(x,v)(u) = \partial^2 p_1(x,v_1)(u) + \partial^2 p_2(x,v_2)(u).$$

• Such tools underlie robust stability characterizations (e.g., tilt stability, Aubin property), and explicit second-order sufficient conditions in composite models and robust statistics.

Recent works elaborate these ideas to prox-regular, nonconvex functions by analyzing f-attentive generalized derivatives, determining the exact bound for variational convexity, and asserting equivalence between strong variational convexity, tilt stability, and strong metric regularity (Gfrerer, 25 Aug 2024).

6. Applications and Expanding Methodologies

Convex and variational analysis is central to state-of-the-art optimization, inverse problems, machine learning, and dynamical systems:

• Spectral variational methods model stability and control for systems governed by spectral abscissa or spectral radius, with explicit subdifferential formulas for nonsmooth optimization over matrices (Burke et al., 2015, Bùi et al., 13 Oct 2025).
• Integral-proximal and composite penalties: Variational analysis of proximal compositions and mixture models provides new convexity-preserving constructions for regularization (e.g., variational Gram functions) and efficient kernelized algorithms (Jalali et al., 2015, Combettes et al., 13 Aug 2024).
• Convexity in infinite dimensions: Variational convexity directly undergirds the design of algorithms in Banach space models (Khanh et al., 2022, Khanh et al., 2023), as well as the proper extension of convexity principles (e.g., Polyak’s ball-image convexity) to uniformly convex Banach settings (Uderzo, 2013).
• Nonlocal and supremal functionals: Notions of Cartesian and separate level convexity determine existence, relaxation, and representation criteria for $L^\infty$-type nonlocal variational problems (Kreisbeck et al., 2022). Smooth approximate convexity of feasible sets provides a geometric bridge between classical convexity and $C^1$-preimage regularity, with implications for path-connectedness and local geometry (Lewis et al., 13 Aug 2024).
• Game theory and equilibria: Convex variational tools provide transparent fixed-point based existence proofs and minimax theorems in infinite-dimensional and nonsmooth contexts, generalizing Nash–von Neumann principles through subdifferential variational calculus (Bao et al., 26 Aug 2024).

In summary, convex and variational analysis has evolved into a mature and richly interconnected discipline, combining geometric, analytic, and algebraic techniques to rigorously address both classical and contemporary problems in nonsmooth, high-dimensional, stochastic, and structured variational models. The field is under active expansion, with ongoing advances in spectral systems, measure-theoretic dualities, infinite-dimensional optimization, and algorithmic variational geometry.