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Maximum theorem

The maximum theorem, also known as Berge's maximum theorem, is a fundamental result in topology and mathematical economics that establishes conditions for the continuity of the maximum value of an objective function and the upper hemicontinuity of the set of its maximizers as parameters vary.^[1] It applies to optimization problems where a continuous function is maximized over a compact-valued and continuous correspondence of feasible sets.^[2] Proved by French mathematician Claude Berge in 1959, the theorem appears in his work Espaces topologiques: fonctions multivoques (translated into English as Topological Spaces in 1963).^[2] In its standard formulation, consider Hausdorff topological spaces X and Y, a continuous real-valued function \phi: Y \to \mathbb{R}, and a nonempty compact-valued continuous correspondence \Gamma: X \rightrightarrows Y; then the value function M(x) = \max \{\phi(y) \mid y \in \Gamma(x)\} is continuous on X, and the argmax correspondence \Phi(x) = \{y \in \Gamma(x) \mid \phi(y) = M(x)\} is nonempty, compact-valued, and upper hemicontinuous.^[2] Upper hemicontinuity here means that for any x_0 \in X and open set V containing \Phi(x_0), there exists a neighborhood U of x_0 such that \Phi(x) \subseteq V for all x \in U, ensuring the maximizers do not "jump" discontinuously.^[1] The theorem's significance lies in its role for comparative statics analysis in economic models, guaranteeing that optimal choices and values respond smoothly to parameter changes, such as shifts in prices or incomes.^[1] For instance, in consumer theory, it implies that the indirect utility function is continuous and the demand correspondence is upper hemicontinuous when preferences are continuous and the budget set is compact and continuous.^[1] Variants extend to weaker conditions, like separate continuity of the objective or different topologies on the action space, broadening its applicability in game theory and dynamic programming.^[1]

Introduction

Statement of the Theorem

Berge's maximum theorem provides conditions under which the value function and the set of maximizers of a parametric optimization problem exhibit continuity properties.^[2] Formally, let X and \Theta be Hausdorff topological spaces, let f: X \times \Theta \to \mathbb{R} be a continuous function, and let C: \Theta \rightrightarrows X be a nonempty compact-valued continuous correspondence. Then the value function f^*(\theta) = \sup_{x \in C(\theta)} f(x, \theta) is continuous on \Theta, and the argmax correspondence C^*(\theta) = \arg\max_{x \in C(\theta)} f(x, \theta) is nonempty, compact-valued, and upper hemicontinuous.^[3]^[2] In this formulation, X serves as the space of choice variables, while \Theta parameterizes the problem, potentially representing exogenous variables or states. The supremum in the value function f^* reflects the optimal attainable value for each \theta, attained over the feasible set C(\theta) due to the compactness of C(\theta) and continuity of f. The argmax correspondence C^* identifies the set of all optimal choices for each \theta.^[3] The theorem relies on three explicit assumptions: the continuity of f with respect to the product topology on X \times \Theta; the compactness of each C(\theta), ensuring the existence of maximizers; and the continuity of C, meaning C is both upper hemicontinuous and lower hemicontinuous as a set-valued map.^[2]^[3]

Historical Context

The Maximum theorem, also known as Berge's maximum theorem, was first proven by the French mathematician Claude Berge in 1959 as part of his work on multivalued functions in topological spaces.^[2] This result appeared in his book Espaces topologiques: fonctions multivoques, which was later translated into English as Topological Spaces and published in 1963.^[2] Berge's theorem established conditions under which the value function of an optimization problem and the set of its maximizers remain continuous with respect to parameters, providing a foundational tool for analyzing parametric optimization in abstract settings. The theorem emerged within the broader development of fixed-point theory in topology during the early to mid-20th century, drawing indirect influences from earlier results on continuity and fixed points. Luitzen Egbertus Jan Brouwer's fixed-point theorem of 1911 guaranteed the existence of fixed points for continuous functions on compact convex sets, laying groundwork for handling continuity in optimization. Shizuo Kakutani's 1941 generalization extended this to upper hemicontinuous set-valued mappings, which proved instrumental for economic applications involving correspondences and helped frame the continuity properties central to Berge's later contribution. These precursors addressed existence but not the parametric stability that Berge formalized, marking a shift toward dynamic and variational analyses in mathematics and economics.^[4] Subsequent work provided rigorous topological clarifications and extensions of Berge's result. Kim C. Border's 1985 monograph Fixed Point Theorems with Applications to Economics and Game Theory offered detailed treatments of the theorem in the context of economic modeling, emphasizing its role in proving equilibrium existence.^[5] Charalambos D. Aliprantis and Kim C. Border further refined these ideas in their 2006 text Infinite Dimensional Analysis: A Hitchhiker's Guide, where they presented the theorem in infinite-dimensional spaces with precise conditions for hemicontinuity and compactness. In applications to Markov decision processes, Eugene A. Feinberg and colleagues in 2013 extended the theorem to noncompact image sets, addressing limitations in dynamic programming by relaxing compactness assumptions while preserving continuity of optimal value functions.^[6] While pre-2013 references, including foundational texts by Border and Aliprantis, continue to dominate discussions of the theorem, recent extensions post-2020 include applications to stochastic dynamic programming and sublinear semigroups in probabilistic settings (as of 2025).^[7]^[8] Berge's original formulation has notably influenced economic theory, enabling analyses of consumer choice and equilibrium under parametric variations.^[9]

Mathematical Preliminaries

Key Definitions

In the context of the maximum theorem, the spaces X and \Theta are topological spaces, which provide the foundational structure for discussing continuity and compactness. A topological space consists of a set X equipped with a collection \mathcal{T} of subsets called open sets, satisfying three axioms: the empty set and X itself are open, the union of any collection of open sets is open, and the finite intersection of open sets is open.^[10] Compactness in a topological space (X, \mathcal{T}) is defined as the property that every open cover— a collection of open sets whose union contains X—admits a finite subcover, meaning a finite subcollection suffices to cover X.^[11] This notion generalizes the Heine-Borel theorem from Euclidean spaces to abstract settings, ensuring that "small" spaces behave well under continuous maps.^[11] A correspondence, also known as a set-valued map, is a function C: \Theta \rightrightarrows X that assigns to each \theta \in \Theta a nonempty subset C(\theta) \subseteq X.^[12] Such maps are compact-valued if, for every \theta \in \Theta, the image C(\theta) is a compact subset of X.^[12] This property ensures that the feasible sets remain "bounded and closed" in a topological sense, facilitating the attainment of maxima. For the objective function f: X \times \Theta \to \mathbb{R}, continuity is a standard requirement. Using the \epsilon-\delta definition, f is continuous at a point (x_0, \theta_0) if, for every \epsilon > 0, there exists \delta > 0 such that whenever \|(x, \theta) - (x_0, \theta_0)\| < \delta, it follows that |f(x, \theta) - f(x_0, \theta_0)| < \epsilon.^[13] Equivalently, in open set terms, f is continuous if the preimage of every open set in \mathbb{R} is open in X \times \Theta.^[10] The value function associated with the optimization problem is defined as f^*(\theta) = \sup_{x \in C(\theta)} f(x, \theta), where the supremum of a set S \subseteq \mathbb{R} is the least upper bound of S, i.e., the smallest number that is greater than or equal to every element in S.^[14] If the supremum is attained, it equals the maximum value of f over C(\theta). The argmax set, or set of maximizers, is given by C^*(\theta) = \{x \in C(\theta) \mid f(x, \theta) = f^*(\theta)\}, which collects all points in the feasible set C(\theta) that achieve the supremum value. This set may be singleton or contain multiple elements, depending on the strictness of the objective function. Upper hemicontinuity of correspondences, a related notion, will be addressed in subsequent sections on continuity types.

Continuity Notions

In topological spaces, the continuity of a single-valued function f: \Theta \to \mathbb{R} can be defined topologically: f is continuous at \theta \in \Theta if for every open neighborhood V of f(\theta), there exists an open neighborhood U of \theta such that f(U) \subseteq V.^[15] Equivalently, via the sequential criterion in metric spaces, f is continuous at \theta if for every sequence \theta_n \to \theta, it holds that f(\theta_n) \to f(\theta).^[15] These definitions ensure that small changes in the parameter \theta lead to small changes in the function value, providing the foundational continuity required for objectives in optimization problems.^[1] For set-valued mappings, known as correspondences C: \Theta \rightrightarrows X, continuity extends to hemicontinuity concepts. Upper hemicontinuity (uhc) at \theta \in \Theta means that for every open set V containing C(\theta), there exists an open neighborhood U of \theta such that C(x) \subseteq V for all x \in U.^[15] This prevents the image set from "exploding" outward under perturbations of \theta.^[1] Lower hemicontinuity (lhc) complements this: for every open set V intersecting C(\theta), there exists an open neighborhood U of \theta such that C(x) \cap V \neq \emptyset for all x \in U, ensuring the image does not "implode" and lose elements.^[15] A correspondence is fully continuous if it is both uhc and lhc.^[1] Compact-valued correspondences play a crucial role in guaranteeing the existence of maxima. A correspondence C is compact-valued if C(\theta) is compact for every \theta \in \Theta.^[15] Under continuity of the objective function, the Extreme Value Theorem implies that a continuous function attains its maximum on each compact set C(\theta), ensuring the argmax set is nonempty.^[1] In the context of the maximum theorem, the argmax correspondence—defined as the set of maximizers of an objective over C(\theta)—inherits upper hemicontinuity from the assumptions of a continuous objective and a compact-valued, uhc constraint correspondence.^[15] This uhc property ensures the stability of solution sets, meaning that as parameters \theta vary continuously, the maximizers do not jump discontinuously to distant points.^[1]

Core Results

The Main Theorem

Under the assumptions of continuity of the objective function f: \Theta \times X \to \mathbb{R} and compactness and continuity of the constraint correspondence C: \Theta \rightrightarrows X, Berge's maximum theorem establishes that the value function f^*(\theta) = \max_{x \in C(\theta)} f(\theta, x) is continuous at every \theta \in \Theta. Furthermore, the optimal solution set C^*(\theta) = \{x \in C(\theta) \mid f(\theta, x) = f^*(\theta)\} is nonempty, as guaranteed by the compactness of C(\theta) and the continuity of f, and compact. Additionally, C^* is upper hemicontinuous, meaning that for any sequence \theta_n \to \theta and x_n \in C^*(\theta_n), there exists a subsequence converging to some x \in C^*(\theta). These conclusions imply that optimal solutions vary continuously with respect to perturbations in the parameters \theta, ensuring stability in optimization problems where parameters may fluctuate slightly. This continuity is particularly vital for comparative statics analysis, as it allows economists and decision theorists to predict how equilibria or optimal choices respond smoothly to changes in exogenous variables, such as prices or endowments, without abrupt discontinuities.^[16] In contrast to standard optimization results for fixed domains, the maximum theorem accommodates set-valued constraints through the use of correspondences, which model situations where feasible sets depend parametrically on \theta and may not be singletons, such as budget sets in consumer theory or strategy sets in games. This framework extends the applicability of continuity guarantees to a broader class of parametric optimization problems prevalent in economic modeling.

Proof Sketch

The proof of the maximum theorem relies on establishing the continuity of the value function f^*(\theta) = \sup_{x \in C(\theta)} f(x, \theta) and the upper hemicontinuity of the argmax correspondence C^*(\theta) = \{x \in C(\theta) : f(x, \theta) = f^*(\theta)\}, under the assumptions that f is continuous and C is compact-valued and continuous.^[1] A key lemma provides the upper semicontinuity of the supremum. For a continuous function f: X \times \Theta \to \mathbb{R} and a compact-valued, upper hemicontinuous correspondence C: \Theta \rightrightarrows X, the function f^*(\theta) = \sup_{x \in C(\theta)} f(x, \theta) is upper semicontinuous.^[2] To show f^* is upper semicontinuous, note that continuity of C implies it is upper hemicontinuous. Combined with the compactness of C(\theta) for each \theta and the lemma above, f^* is upper semicontinuous.^[1]^[2] For lower semicontinuity of f^*, proceed by contradiction. Suppose there exists a sequence \theta_n \to \theta such that f^*(\theta_n) \to l < f^*(\theta). Let x^* \in C(\theta) be a maximizer, so f(x^*, \theta) = f^*(\theta). By lower hemicontinuity of C, there exist x_n \in C(\theta_n) with x_n \to x^*. Continuity of f then implies f(x_n, \theta_n) \to f(x^*, \theta) = f^*(\theta) > l. However, f(x_n, \theta_n) \leq f^*(\theta_n) \to l, yielding a contradiction. Thus, f^* is lower semicontinuous.^[17]^[1] The argmax correspondence C^* is nonempty for each \theta by the extreme value theorem, as f is continuous on the compact set C(\theta).^[1] C^*(\theta) is compact, being a closed subset of the compact set C(\theta).^[18] It is upper hemicontinuous: for \theta_n \to \theta and x_n \in C^*(\theta_n), compactness yields a convergent subsequence x_{n_k} \to x \in C(\theta); upper semicontinuity of f^* and continuity of f ensure x \in C^*(\theta).^[18]^[1]

Extensions

Minimization Variant

The minimization variant of Berge's maximum theorem addresses optimization problems where the objective is to minimize a continuous function subject to parameter-dependent constraints. Specifically, let f: X \times \Theta \to \mathbb{R} be a continuous function, where X and \Theta are topological spaces, and let C: \Theta \rightrightarrows X be a nonempty, compact-valued, continuous correspondence. Then, the minimal value function f_*(\theta) = \inf_{x \in C(\theta)} f(x, \theta) is continuous at every \theta \in \Theta, and the argmin correspondence C_*(\theta) = \arg\min_{x \in C(\theta)} f(x, \theta) is nonempty, compact-valued, and upper hemicontinuous.^[19]^[20] This result follows directly from the main theorem by applying it to the negated objective function -f, which inherits continuity from f. The supremum of -f over C(\theta) equals -f_*(\theta), so the continuity of the maximal value for -f implies the continuity of f_*. Likewise, the argmax set for -f coincides with the argmin set for f, and upper hemicontinuity is preserved under this identification.^[20] The minimization variant finds frequent application in constrained optimization settings, such as dynamic programming models where agents minimize costs or losses over parameter-dependent feasible sets.^[21]

Generalizations under Convexity

One prominent generalization of Berge's maximum theorem incorporates quasiconcavity of the objective function f alongside upper semicontinuity, relaxing the full continuity requirement while preserving desirable properties of the argmax correspondence C^*. Specifically, if f is upper semicontinuous and quasiconcave in the choice variable x for each parameter \theta, and the constraint correspondence C is continuous with convex values, then C^* remains convex-valued and upper hemicontinuous (uhc).^[22] This extension ensures that the set of maximizers forms a convex set, which is particularly useful in economic models where preferences exhibit quasiconcavity, such as in consumer theory with non-convex budget sets.^[23] Further refinements arise when f is concave and C is convex-valued. Under these conditions, C^* is convex-valued, and if f is strictly concave in x, then C^* becomes single-valued, implying a unique maximizer for each \theta.^[22] These results stem from the interaction between the concavity of f, which guarantees convexity of upper level sets, and the convexity of C(\theta), ensuring that the maximum is attained over a convex domain without multiplicity in the strict case.^[23] Such generalizations enhance the theorem's applicability to optimization problems with convex structures, like linear programming relaxations or concave utility maximization. In infinite-dimensional settings, such as Banach spaces, these convexity-based extensions adapt to weaker topologies, like the weak topology, to handle non-compact domains. Aliprantis and Border provide a framework where Berge's theorem holds for correspondences between locally convex topological vector spaces, with convexity assumptions ensuring upper hemicontinuity of C^* even under weak convergence, which is crucial for functional analysis in economics and control theory.^[24] Recent developments address stochastic environments with non-compact sets, particularly in Markov decision processes (MDPs). Bertsekas extends the theorem to undiscounted stochastic optimal control problems over non-compact action spaces, using convexity of level sets and growth conditions on the objective to guarantee convergence of value functions and policy iterations, filling gaps in earlier compact-set assumptions post-2013.^[25]

Applications and Examples

Economic Applications

In consumer theory, the Maximum Theorem plays a pivotal role in establishing the continuity of the demand correspondence, which maps prices and income to the set of utility-maximizing consumption bundles over the budget set. Specifically, when the utility function is continuous and the budget correspondence is continuous (as it is for non-negative prices and income), the theorem guarantees that the demand correspondence is upper hemicontinuous, ensuring stable consumer behavior under small perturbations in economic parameters. This property is essential for deriving comparative statics results, such as how demand responds predictably to price changes without discontinuous jumps.^[26] In general equilibrium theory, the theorem underpins the continuity of excess demand correspondences in the Arrow-Debreu model, facilitating the application of fixed-point theorems like Brouwer's or Kakutani's to prove the existence of competitive equilibria. By ensuring that individual demand functions are continuous in prices, the Maximum Theorem allows aggregation to economy-wide excess demands that satisfy the conditions for equilibrium existence, even in multi-commodity settings with production. This linkage is central to the model's robustness, as it validates equilibria under varying endowments and technologies without requiring additional convexity assumptions beyond those for individual optimization.^[27] In game theory, the Maximum Theorem ensures the upper hemicontinuity of best-response correspondences in normal-form games, where payoffs are continuous in strategies, thereby supporting the continuity of Nash equilibrium sets under perturbations to payoffs or strategy spaces. This stability is crucial for refining existence proofs via fixed-point methods and analyzing equilibrium selection in games with multiple solutions.^[27] For instance, it implies that small changes in players' utilities lead to nearby equilibria, aiding comparative statics in strategic interactions. Applications to behavioral economics remain limited, with extensions to non-expected utility models like prospect theory maximizers appearing primarily in post-2015 works that adapt the theorem to kinked or discontinuous value functions while preserving continuity of choice sets.^[28] Recent applications as of 2025 extend the theorem beyond traditional economics. In AI alignment, it ensures the upper hemicontinuity of optimal reward model selections under perturbations in human feedback parameters, supporting stable training of large language models.^[29] In control theory, it analyzes the continuity of feedback controllers defined by parametric optimizations with multiple constraints, aiding robust design in engineering systems.^[30]

Illustrative Examples

A simple example of the maximum theorem involves quadratic utility maximization subject to an interval constraint. Consider the objective function f(x, \theta) = -(x - \theta)^2, where x is the choice variable and \theta \in \mathbb{R} is the parameter, maximized over the constraint set C(\theta) = [0,1] for all \theta. The unconstrained maximizer is x = \theta, so the constrained argmax correspondence is C^*(\theta) = \{\mathrm{proj}_{[0,1]}(\theta)\}, where \mathrm{proj}_{[0,1]}(\theta) = 0 if \theta < 0, \theta if $0 \leq \theta \leq 1, and $1 if \theta > 1. This single-valued correspondence is continuous, hence upper hemicontinuous, with jumps in the sense of boundary behavior at \theta = 0 and \theta = 1 where the maximizer snaps to the endpoint. The value function is f^*(\theta) = \max_{x \in [0,1]} -(x - \theta)^2 = -(\mathrm{proj}_{[0,1]}(\theta) - \theta)^2, which equals 0 for \theta \in [0,1], -\theta^2 for \theta < 0, and -(\theta - 1)^2 for \theta > 1; this is continuous in \theta as the distance to the interval squared is continuous. These properties follow from the continuity of f and the constant compact-valued continuous constraint correspondence C. For the minimization variant, consider a production cost minimization problem with linear cost function f(x, w) = w x, where w > 0 is the wage parameter and x \geq 0 is the output level, minimized over the linear constraint set C(w) = [a(w), b(w)] with a(w) = 0 and b(w) = k w for some k > 0. Since the objective is increasing in x for fixed w > 0, the minimizer is at the lower bound, so the argmin correspondence is C_*(w) = \{a(w)\} = \{0\}, which is single-valued and continuous, hence upper hemicontinuous. The infimum value is f_*(w) = w \cdot a(w) = 0, which is continuous in w. Applying the minimization variant of the theorem, with f continuous and C compact-valued and continuous (as linear functions are continuous), ensures the continuity of f_* and upper hemicontinuity of C_*. The theorem's assumptions are crucial, as failures occur without them; for instance, a discontinuous constraint set can lead to a discontinuous argmax correspondence. Consider f(x, \theta) = -x^2 maximized over C(\theta) = [0,1] if \theta < 0.5 and C(\theta) = [2,3] if \theta \geq 0.5, a closed-valued but discontinuous correspondence at \theta = 0.5. The argmax is C^*(\theta) = \{0\} for \theta < 0.5 and C^*(\theta) = \{2\} for \theta \geq 0.5, which jumps discontinuously at \theta = 0.5 and is not upper hemicontinuous there, since sequences approaching from below have limit points not in C^*(0.5). The value function f^*(\theta) = 0 for \theta < 0.5 and f^*(\theta) = -4 for \theta \geq 0.5 is also discontinuous. This contrasts with the theorem's requirements, as C fails continuity. To illustrate in a dynamic setting, consider a simple infinite-horizon Markov decision process (MDP) for inventory control with backorders. The state x \in \mathbb{R} is the inventory level, action a \in \mathbb{R}^+ is the order quantity, demand D is i.i.d. with finite support, and dynamics are x' = x + a - D. The per-period cost is c(x, a) = K \cdot \mathbf{1}_{\{a > 0\}} + \bar{c} a + \mathbb{E}[h(x + a - D)], where K \geq 0, \bar{c} > 0, and h is convex and continuous representing holding/backorder costs. The value function v_\alpha(x) = \min_a \{ c(x, a) + \alpha \mathbb{E}[v_\alpha(x + a - D)] \} for discount factor \alpha \in (0,1) is continuous in x, with the argmin correspondence upper hemicontinuous, by applying Berge's maximum theorem (via the minimization variant) to the Bellman operator, given the continuity of costs and transitions. Optimal policies are of the (s, S) form under mild conditions on h, ensuring computable continuous value functions for planning.^[31]

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