Strict Minimax Inequality Analysis

• Strict minimax inequality is defined by the strict ordering between infimum–supremum and supremum–infimum values when classical convexity or compactness assumptions are relaxed.
• It extends classical minimax formulations to settings with generalized convexity, coupled constraints, and noncompact domains, thereby uncovering deeper insights in variational analysis.
• This principle is applied to establish existence, uniqueness, and multiplicity of solutions in critical point theory, PDE problems, and optimization models.

The strict minimax inequality refers to settings in which the classical minimax equality $\inf_{x}\sup_{y}f(x,y)=\sup_{y}\inf_{x}f(x,y)$ fails and, instead, one obtains a strict ordering, usually $\inf_{x}\sup_{y}f(x,y)<\sup_{y}\inf_{x}g(x,y)$ or analogous formulations in the presence of constraints, generalized convexity, or noncompactness. The phenomenon arises in a diverse range of functional, variational, and optimization contexts, often yielding deeper existence, uniqueness, multiplicity, or qualitative properties in critical point theory and saddle-point analysis.

1. Minimax Formulations: Classical and Strict Variants

The classical minimax principle asserts, under convexity and compactness-type conditions on functionals $f(x, y)$ defined over suitable sets $X,Y$, that

$$\inf_{x \in X} \sup_{y \in Y} f(x, y) = \sup_{y \in Y} \inf_{x \in X} f(x, y).$$

The strict minimax inequality arises whenever the hypotheses are relaxed or generalized in such a way that the equality becomes a strict inequality. Scenarios include:

• Systems of coupled constraints where minimization and maximization variables interact non-trivially, often through linear constraints $A x + B y \le c$ (Tsaknakis et al., 2021).
• Function families $f,g$ with strict pointwise gaps and strictly generalized convexity or concavity properties, such that $\inf_X\sup_Y f < \sup_Y\inf_X g$ (Bachir, 24 Jun 2024).
• Multi-level variational problems, e.g., higher eigenvalue characterizations, where the critical level is strictly beneath a natural threshold (Perera et al., 2012).
• Abstract settings employing penalty or perturbation schemes to force separation between the two sides of the minimax relationship (Ricceri, 2012).

2. Generalized Convexity and Simons–Like Inequalities

Recent theory extends classical minimax theorems to generalized convexlike and concavelike settings. Key concepts include:

• $t$–convexlike functions: $f:X\times Y\to\mathbb{R}$ is $t$–convexlike on $X$ if, for all $x_1,x_2\in X$, there exists $x_3\in X$ satisfying $f(x_3,y)\le t f(x_1,y)+(1-t)f(x_2,y)$ for every $y\in Y$.
• infsup–convexity and supinf–concavity: These generalize convexity/concavity by integrating infimum-superior and supremum-inferior operations over convex combinations of function evaluations, as precisely defined in (Bachir, 24 Jun 2024).
• Simons–like inequalities: For all nets $(x_\alpha)\subset X$,

$$\inf_{x \in X}\sup_{y \in Y}f(x,y)\le \sup_{y \in Y} \limsup_{\alpha} f(x_\alpha, y)$$

plays a central role in deducing minimax inequalities and their strict counterparts. A strict variant is standard in the presence of strict convexity/concavity analogues and pointwise strict gaps $f<g$, producing strict inequalities $\inf_{X}\sup_{Y}f<\sup_{Y}\inf_{X}g$ (Bachir, 24 Jun 2024).

3. Strict Minimax Criterion: Abstract Presentations and Applications

Ricceri's strict minimax criterion introduces a flexible abstract framework based on functional–topological data sets $(X,A,Y,N,p,\Psi,J)$ (Ricceri, 2012). The main theorem states that, under appropriate semicontinuity, injectivity, and non-constancy conditions (see definitions of $\mathcal G$, $\mathcal H$, and $\mathcal M$ classes), and provided a gap parameter $p>\Theta$ (quantifying the strict separation),

$$\sup_{x\in E}\inf_{a\in A}\left(J(x)-p\,p(\Psi(x,a))\right) < \inf_{a\in A}\sup_{x\in E}\left(J(x)-p\,p(\Psi(x,a))\right)$$

for some $E$ in any filtering cover $N$ of $X$. The strict minimax criterion is then exploited to establish existence and uniqueness of minimizers, non-existence results in constrained spaces, critical point multiplicity, and penalty function characterizations. For $C^1$–functions $J$ with convex perturbations $\varphi$, strict minimax conditions guarantee not only existence but uniqueness of global minimizers, and explicit bounds on the gap between minimum values and perturbed functionals (see Section 3 and Theorem 1.3 in (Ricceri, 2012)).

4. Duality, Coupled Constraints, and Computational Complexity

The strict minimax inequality is particularly prominent in minimax problems with linearly coupled constraints. For the problem

$$\min_{x\in X} \max_{y\in Y,\,A x + B y \le c} f(x,y)$$

the classical ordering between $\max \min$ and $\min \max$ can fail, often yielding

$V_{\max\min} < V_{\min\max}$

as shown in explicit quadratic examples in (Tsaknakis et al., 2021). This strict gap arises when the coupling constraint $A x + B y \le c$ makes the feasible set for $y$ depend on $x$, thereby disrupting the interchangeability of $\max$ and $\min$ even under convexity–concavity assumptions. The duality framework employs Lagrangian multipliers and yields weak and strong duality theorems. Strong duality (i.e., zero duality gap) can be recovered under additional conditions (e.g., strong concavity in $y$ and existence of Slater points), but generic coupled minimax problems remain NP-hard and nonconvex, with the strict minimax gap quantifying a defender–attacker asymmetry in network and game-theoretic models (Tsaknakis et al., 2021).

5. Variational Analogues: Second Minimax Level and Strictness

In nonlinear PDE and variational analysis, strict minimax inequalities appear in eigenvalue problems at higher critical levels. For the scalar field equation on $\mathbb{R}^N$ with energy functional $J(u)$ and mass constraint $I(u)=1$,

$$\lambda_2 = \inf_{\gamma \in \Gamma_2} \max_{y \in S^1} J(\gamma(y))$$

is the minimax characterization of the second eigenvalue. A threshold $\lambda^\#$ is defined such that always $\lambda_2 \le \lambda^\#$, with strict inequality $\lambda_2 < \lambda^\#$ ensured by introducing a potential penalty $V_\infty - V(x)\ge c e^{-a|x|}$ near infinity for suitable $(c,a)$ (Perera et al., 2012). This strict gap is crucial for demonstrating the existence of critical points at the second level, enforcing compactness properties (via concentration–compactness) and providing qualitative features such as nodality and symmetry breaking.

6. Noncompact, Nonlinear, and Inf-Convolution Examples

The strict minimax phenomenon persists beyond classical compact settings. New minimax theorems valid in noncompact situations affirm that strict versions can always be constructed by enforcing strictness in the generalized convexity or concavity definitions, along with strict pointwise separation between function families (Bachir, 24 Jun 2024). Notable cases include:

• Functions governed by control metrics $\theta(x_1,x_2)$ imposing $t$–convexlike properties and Simons–like inequalities, leading to strict minimax gaps when one function is strictly dominated by another at some point.
• Inf-convolution constructions of the form

$$f(x,y) = \inf_{z \in X} \{ g(z, y) + K\, \theta(x, z) \}$$

where $g$ is strictly $s$–concavelike in $y$; strict minimax inequalities arise when there is a strict gap embedded in the inf-convolution term.

7. Summary Table: Domains of Strict Minimax Inequality

Setting	Mechanism for Strictness	Main Consequence/Use
Generalized convexity	Strict $t$–convexlike/supinf–concave, $f<g$	Strict ordering in minimax values
Coupled linear constraints	Feasible region depends on $x$	Duality gap, NP-hardness, attacker/defender asymmetry
Variational/PDE problems	Potential penalties, symmetry breaking	Existence/multiplicity of nodal/nonradial solutions
Penalty frameworks	Strict separation of perturbed minimizers	Uniqueness and location results

The strict minimax inequality thus encapsulates a broad principle by which classical saddle-point symmetry is replaced by strict separation, yielding new insight into variational structure, game theory, and nonlinear analysis (Bachir, 24 Jun 2024, Ricceri, 2012, Tsaknakis et al., 2021, Perera et al., 2012).