LP Duality-Based Characterization

• LP duality-based characterization is a framework that uses duality to analyze optimality and feasibility in linear programs via their associated dual problems.
• It establishes a structural equivalence between LP formulations and zero-sum game equilibria, facilitating certificate generation and efficient solution methods.
• The approach enables cross-domain applications in operations research, machine learning, and economics by leveraging robust algorithms and duality insights.

Linear programming (LP) duality-based characterization refers to the rigorous framework by which the optimization, feasibility, and solution structure of linear programs are understood via their dual problems, often in profound connection with other mathematical disciplines such as game theory, combinatorial optimization, coding theory, and more. The archetype of such characterization is strong duality: for every linear program (LP), there exists an associated dual LP, and under appropriate conditions (most commonly feasibility and boundedness), their optimal values coincide and the structure of their solutions is tightly interrelated. In recent research, these duality correspondences have been exploited both to transfer results between domains and to derive new constructive or certificate-generating proofs of fundamental theorems in optimization and mathematical game theory, as exemplified in "Zero-Sum Games and Linear Programming Duality" (Stengel, 2022).

1. LP Duality and Zero-Sum Game Theory

A foundational link exists between the strong duality theorem for LP and the minimax theorem in zero-sum games. For a matrix game with payoff matrix $A$, the max-min problem for Player 1

$\max_{x \in X} \min_{y \in Y} x^\top A y$

can be formulated as a linear program, where $X$ and $Y$ are probability simplices enforcing mixed strategies. The LP for Player 1 is

$$\begin{align*} \text{maximize} &~v \ \text{subject to} &~x^\top A \geq v \mathbf{1},~\sum_{i} x_{i} = 1,~x_{i} \geq 0 \end{align*}$$

and the dual for Player 2 is

$$\begin{align*} \text{minimize} &~w \ \text{subject to} &~A y \leq w \mathbf{1},~\sum_{j} y_{j} = 1,~y_{j} \geq 0 \end{align*}$$

The strong duality theorem assures that, whenever both programs are feasible, their optimal values coincide: the value of the game equals the optimal value of both primal and dual LPs. This equivalence is not merely formal—zero-sum games and LP are structurally isomorphic, with strategies corresponding to feasible points in their respective polytopes, and equilibrium value equating to optimal LP objective value. Feasible solutions to the LPs correspond precisely to optimal mixed strategies in the associated game.

2. Formal Equivalence and Constructive Duality

The minimax theorem

$$\max_{x \in X} \min_{y \in Y} x^\top A y = \min_{y \in Y} \max_{x \in X} x^\top A y$$

is shown to be formally equivalent to LP strong duality. Classical proofs usually proceed by showing that LP strong duality implies the minimax theorem, but the converse is nontrivial. The cited paper provides a constructive proof that begins with the minimax equilibrium existence and yields the LP strong duality property, circumventing incomplete steps in traditional derivations (notably Dantzig 1951). Specifically, it interprets every LP as a zero-sum game between a maximizer and minimizer over convex sets, leveraging the minimax theorem to affirm the existence and equality of optimal values and solutions.

3. Certificates, Infeasibility, and Extending Dantzig's Construction

A novel aspect is the extension of Dantzig’s "game", in which the correspondence between LP and matrix games is made explicit and robust. The central result is that for any max-min strategy (not just optimal), there is an algorithmic path:

• If the strategy yields a finite game value, it generates a feasible, optimal solution to the LP.
• If not, it produces a certificate of infeasibility (such as a separating hyperplane or a witness to unboundedness).

Thus, attempts to optimize within the zero-sum game framework yield not only LP optima, but also information about infeasibility where appropriate.

4. Tradeoffs and Generality of Characterization

The duality-based characterization framework exhibits several important tradeoffs:

• Generalization: Any finite LP can be interpreted as a zero-sum game, and vice versa.
• Feasibility and Boundness: Duality gaps precisely correspond to infeasibility or unboundedness in either the primal or dual, which in game-theoretic parlance manifest as the absence of equilibria.
• Certificate Strength and Algorithmic Implications: A feasible solution for either LP provides optimal strategies for both players, while infeasibility produces strong separation certificates, akin to Farkas lemma witnesses.

In particular, the simplex method's geometric representation of LP solutions aligns with the polyhedral structure of mixed strategy sets.

5. Implementation, Scaling, and Applications

The duality-based characterization is highly practical owing to LP's well-understood scaling properties and the existence of efficient solution algorithms (e.g., simplex, interior-point, ellipsoid methods). The equivalence to zero-sum games allows for cross-fertilization of algorithms—e.g., primal-dual methods, combinatorial pivoting, and polynomial-time LP solvers can be applied to matrix games and vice versa. Moreover, game-theoretic interpretation of LP certificates facilitates the design of separation oracles, infeasibility certificates, and verification proofs in mathematical programming.

In real-world domains:

• Operations research: Resource allocation and scheduling can be framed in game terms, with dual certificates providing sensitivity and shadow price information.
• Machine learning: Adversarial learning problems (minimax optimization) correspond directly to primal-dual LPs.
• Economic theory: Market equilibria for linear utility models naturally instantiate primal-dual LPs as their game-theoretic analogs.

The above equivalence enables practitioners to select optimal implementation approaches per instance, balancing the expressiveness of game-theoretic modeling with the scalability of modern LP solvers.

6. Broader Context and Further Directions

The duality-based characterization animates further research bridging optimization, algorithmic game theory, and computational mathematics. It lays the groundwork for:

• Generalized duality frameworks: Extensions to conic programs, convex optimization, and infinite games.
• Algorithmic robustness: Failure of strong duality directly signals pathologies in model formulation, guiding diagnosis and repair.
• Certificate-based verification: LP duality certifies solutions, optimality, and infeasibility in a way that can be checked independently or embedded in algorithmic solvers.

This comprehensive perspective, encompassing classical minimax theory and modern LP duality, organizes and unifies broad classes of optimization and equilibrium computations across mathematical disciplines.

Key Table: LP–Game Duality Correspondence

Linear Program	Zero-Sum Game	Duality/Equilibrium
Primal: $\max\{c^\top x : A x \leq b, x \geq 0\}$	Player 1: $\max_{x \in X}\min_{y \in Y} x^\top A y$	Primal optimum is game value
Dual: $\min\{b^\top y : A^\top y \geq c, y \geq 0\}$	Player 2: $\min_{y \in Y}\max_{x \in X} x^\top A y$	Dual optimum is same game value
Nonzero gap/infeasibility	No equilibrium (unbounded/infeasible)	Certificate of infeasibility/unboundedness

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LP duality-based characterization thus provides both the theoretical bedrock and practical pathways for analyzing and solving linear programs and zero-sum games, with strong implications for certificate-generation, solution structure, and robustness in optimization models (Stengel, 2022).