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Concave function

In mathematics, a concave function is a function f: S \to \mathbb{R}, defined on a convex set S in a vector space, that satisfies the inequality f(tx + (1-t)y) \geq t f(x) + (1-t) f(y) for all points x, y \in S and all scalars t \in [0,1].^[1] This condition ensures that the graph of f lies above or on any chord (straight line segment) connecting two points on the graph, providing a geometric interpretation of the function's curvature in the sense of supporting lines from below.^[1] Concave functions exhibit several important properties that distinguish them from other function classes. A function f is concave if and only if its negative -f is convex, establishing a direct duality between the two concepts.^[1] For twice continuously differentiable functions, concavity holds if and only if the Hessian matrix D^2 f(x) is negative semidefinite at every point x in the domain, meaning all eigenvalues of the Hessian are less than or equal to zero.^[1] Additionally, the superlevel sets \{ x \in S \mid f(x) \geq \alpha \} of a concave function are convex sets, which facilitates analysis in geometric and optimization contexts.^[2] Concave functions play a central role in optimization theory, where maximizing a concave objective function over a convex feasible set guarantees that any local maximum is a global maximum, simplifying the solution of such problems via first-order conditions.^[2] In economics, they model key behaviors such as diminishing marginal utility in consumer preference functions, where the marginal rate of substitution decreases as consumption increases, and diminishing marginal returns in production functions.^[2] These applications extend to broader fields like operations research and machine learning, where concavity ensures well-behaved algorithms for problems involving resource allocation and risk assessment.^[1]

Definition and Interpretation

Formal Definition

A function f: X \to \mathbb{R}, where X is a convex subset of \mathbb{R}^n, is concave if for all x, y \in X and \lambda \in [0,1],

f(\lambda x + (1-\lambda)y) \geq \lambda f(x) + (1-\lambda) f(y).

^[3] This inequality expresses the idea that the function value at any convex combination of points lies above the corresponding convex combination of function values, ensuring the graph of the function bends "upward" relative to the chord connecting any two points in the domain.^[4] An equivalent formulation is that f is concave if and only if -f is convex.^[3] This duality highlights the symmetry between concavity and convexity in optimization theory. The domain X must be convex to guarantee that convex combinations remain within X, and the function is typically real-valued over vector spaces like \mathbb{R}^n.^[5] Affine functions, of the form h(x) = a^\top x + b for a \in \mathbb{R}^n and b \in \mathbb{R}, satisfy both the concave and convex inequalities with equality and thus represent the boundary case between the two classes.^[3]

Geometric Interpretation

A concave function exhibits a distinctive geometric shape in its graph over a convex domain. Unlike the graph of a convex function, which lies below any chord connecting two points on it, the graph of a concave function lies above such chords, meaning the secant line segment connecting any two points on the graph falls entirely below or on the graph itself. This property visually distinguishes concavity as a form of "upward bowing" or dished curvature, where the function values between points exceed the straight-line connection, evoking an inverted bowl or saucer-like appearance in plots.^[6] The hypograph of a concave function f, defined as the set \{(x, t) \mid x \in \dom f, \, t \leq f(x)\} consisting of all points below or on the graph, forms a convex set in the product space.^[6] This convexity of the hypograph provides a foundational geometric characterization equivalent to the function's concavity, mirroring how the epigraph characterizes convexity for the opposite case. Geometrically, the chord condition illustrates this through linear interpolation: for any distinct points x and y in the domain and \theta \in (0,1), the value f(\theta x + (1-\theta) y) lies at or above the interpolated value \theta f(x) + (1-\theta) f(y), ensuring the graph arches above the secant line.^[7] This intuitive bending reinforces the function's tendency toward higher intermediate values relative to straight-line approximations, underscoring its role in capturing peaking or diminishing behaviors in graphical representations.^[8]

Properties and Characterizations

Univariate Case

In the univariate case, a function f: \mathbb{R} \to \mathbb{R} is concave if it satisfies the general definition of concavity, meaning the graph lies above any chord connecting two points in its domain.^[3] For differentiable functions, a first-order condition characterizes concavity: f is concave if and only if its derivative f' is nonincreasing on the domain, which implies that for all x, y in the domain with x < y, f'(y) \leq f'(x).^[3] Equivalently, the function lies below its tangent lines, satisfying f(y) \leq f(x) + f'(x)(y - x) for all x, y in the domain.^[3] For twice continuously differentiable functions f: \mathbb{R} \to \mathbb{R}, the second derivative test provides a simple criterion: f is concave if and only if f''(x) \leq 0 for all x in the domain.^[3] This condition ensures the graph curves downward or is linear, reflecting the nonincreasing nature of f'. A key inequality for concave functions is Jensen's inequality, which states that for weights \lambda_i \geq 0 with \sum \lambda_i = 1 and points x_i in the domain,

f\left( \sum \lambda_i x_i \right) \geq \sum \lambda_i f(x_i).

This follows directly from the definition and extends to expectations: if X is a random variable with existing moments, then f(\mathbb{E}[X]) \geq \mathbb{E}[f(X)].^[3] Concave functions on \mathbb{R} exhibit specific monotonicity behavior: they are either nonincreasing (f'(x) \leq 0 for all x), nondecreasing (f'(x) \geq 0 for all x), or achieve a global maximum at some point c, being nondecreasing on (-\infty, c] and nonincreasing on [c, \infty).^[3] Concavity is preserved under certain compositions: if g is concave and nondecreasing, and f is concave, then the composition g \circ f is concave.^[3] Additionally, composition with affine functions, such as f(Ax + b) where A is a scalar and b \in \mathbb{R}, maintains concavity.^[3]

Multivariate Case

In the multivariate setting, a twice continuously differentiable function f: \mathbb{R}^n \to \mathbb{R} defined on a convex domain is concave if and only if its Hessian matrix \nabla^2 f(x) is negative semidefinite at every point x in the domain.^[6] This condition means that for all vectors v \in \mathbb{R}^n and all x in the domain, v^T \nabla^2 f(x) v \leq 0.^[6] The negative semidefiniteness of the Hessian ensures that the function does not curve upward in any direction, generalizing the univariate second-derivative test to higher dimensions.^[6] Jensen's inequality extends to multivariate concave functions as follows: for a concave function f and a probability measure \mu on a convex set in \mathbb{R}^n, the integral satisfies \int f \, d\mu \geq f\left( \int x \, d\mu \right).^[6] This formulation captures the expectation-based version where, for a random vector X with distribution \mu, f(\mathbb{E}[X]) \geq \mathbb{E}[f(X)], provided X lies in the domain almost surely.^[6] The inequality reflects the function's tendency to lie above its chords, even under general probability measures beyond finite convex combinations.^[6] Concave functions are quasi-concave, meaning their superlevel sets \{x \mid f(x) \geq \alpha\} are convex for every \alpha \in \mathbb{R}.^[6] However, the converse does not hold; quasi-concave functions may fail to satisfy the full concavity inequality, as their upper-level sets are convex without requiring the function to lie above all secant lines.^[9] This distinction arises because quasi-concavity only demands convexity of superlevel sets, allowing for functions that are "bowl-shaped" in a weaker sense than full concavity.^[9] Certain operations preserve concavity for multivariate functions. Specifically, non-negative linear combinations of concave functions—such as \sum \alpha_i f_i where \alpha_i \geq 0 and \sum \alpha_i = 1—remain concave, as they correspond to convex combinations of the original functions.^[6] The pointwise infimum of a family of concave functions is concave, since the hypograph of the infimum is the intersection of the hypographs of the individual functions, which are convex sets.^[6] In contrast, the pointwise supremum of a family of concave functions is not necessarily concave, as the hypograph of the supremum is the union of the individual hypographs, which is not necessarily convex.^[6] For differentiability in the multivariate case, concave functions defined on open convex domains in \mathbb{R}^n are analyzed using Gâteaux and Fréchet derivatives. A concave function on such a domain is Gâteaux differentiable at an interior point x if the directional derivative exists in every direction h, given by \lim_{t \to 0} \frac{f(x + t h) - f(x)}{t}.^[10] If Gâteaux differentiability holds at x, then the function is necessarily Fréchet differentiable there, meaning the derivative is linear and continuous in the direction, satisfying f(x + h) = f(x) + \nabla f(x)^T h + o(\|h\|) as \|h\| \to 0. This equivalence stems from the sublinear growth bound imposed by concavity on convex domains.^[10]

Relation to Convexity

Equivalence with Convex Functions

A function f defined on a convex set is concave if and only if its negation -f is convex. This equivalence follows directly from the definitions: the concavity condition f(\lambda x + (1-\lambda)y) \geq \lambda f(x) + (1-\lambda) f(y) for \lambda \in [0,1] and x, y in the domain negates to -f(\lambda x + (1-\lambda)y) \leq \lambda (-f(x)) + (1-\lambda) (-f(y)), which is precisely the convexity inequality for -f. Conversely, if -f is convex, negating the convexity inequality recovers the concavity of f. This relationship has significant implications in optimization, where maximizing a concave objective function over a convex set is equivalent to minimizing the convex function -f over the same set, allowing the application of convex optimization techniques to concave maximization problems. In convex analysis, concave functions play a key role in duality theory, particularly in Lagrangian duality, where the dual problem involves maximizing a concave perturbation of the primal objective to derive strong duality bounds and optimality conditions. Borderline cases highlight the symmetry: affine functions, which satisfy equality in the convexity inequality, are both convex and concave, as are their negations. A function is strictly concave if and only if its negation is strictly convex, meaning the inequality is strict for \lambda \in (0,1) and x \neq y. The terminology for concave functions evolved alongside convex analysis in the early 20th century, with Johan Jensen's 1906 work on convex functions and Jensen's inequality establishing foundational concepts that directly influenced the definition of concavity as the negation of convexity.

Strict and Strong Concavity

A strictly concave function satisfies a strengthened version of the concavity inequality, where the function value at any convex combination of distinct points exceeds the corresponding convex combination of the function values. Formally, a function f defined on a convex set is strictly concave if, for all x, y in the domain with x \neq y and all \lambda \in (0,1),

f(\lambda x + (1-\lambda) y) > \lambda f(x) + (1-\lambda) f(y).

This strict inequality ensures that the graph of f lies strictly above the line segment connecting any two points on the graph, precluding flat segments where equality would hold.^[6] Strong concavity imposes an even stronger condition by quantifying the deviation above the chord with a quadratic term. A function f is strongly concave with modulus m > 0 if, for all x, y in the domain and all \lambda \in (0,1),

f(\lambda x + (1-\lambda) y) \geq \lambda f(x) + (1-\lambda) f(y) + \frac{m}{2} \lambda (1-\lambda) \|x - y\|^2.

This additional positive term guarantees quadratic growth in the amount by which the function exceeds the chord, providing a uniform measure of "curvature" that distinguishes it from mere strict concavity.^[6] For twice-differentiable functions, strict concavity is characterized by the Hessian matrix being negative definite everywhere in the domain, meaning \nabla^2 f(x) \prec 0 for all x, which ensures the local second-order approximation is strictly concave. Strongly concave functions, being a subclass of strictly concave ones, inherit this property but with the stronger Hessian condition \nabla^2 f(x) \preceq -m I for some m > 0, implying unique global maximizers over convex sets when they exist. In contrast to non-strict concavity, strict concavity eliminates any linear segments in the function, while strong concavity's quadratic term ensures a bounded rate of deviation that aids in convergence analyses. The negation of a strictly (respectively, strongly) concave function is strictly (strongly) convex.^[6]^[1]^[11]

Examples

Basic Examples

Constant functions, such as f(x) = c where c is a constant, are trivially concave, as they satisfy the concavity inequality with equality for all points in their domain.^[12] These functions have a zero second derivative, confirming their concavity everywhere.^[6] Linear functions of the form f(x) = ax + b, where a and b are real constants, are concave (and also convex) over the entire real line, since affine functions preserve convex combinations exactly.^[13] Their second derivative is zero, aligning with the second derivative test for concavity in the univariate case.^[6] A classic example of a strictly concave quadratic function is f(x) = -x^2, whose graph forms a downward-opening parabola.^[6] The second derivative f''(x) = -2 < 0 for all x demonstrates its strict concavity across the real numbers.^[13] The natural logarithmic function f(x) = \log x (using the natural base) is concave on the positive real numbers x > 0.^[6] This follows from its second derivative f''(x) = -1/x^2 < 0, which is negative everywhere in the domain.^[14] Piecewise linear functions can also be concave if their slopes are non-increasing across segments. For instance, the function defined as f(x) = x for $0 \leq x \leq 0.5 and f(x) = 1 - x for $0.5 \leq x \leq 1 (often called an inverted tent function) has slopes 1 and then -1, satisfying the condition for concavity on [0,1].^[13] Such functions, with weakly decreasing slopes, are concave by the characterization for univariate cases.^[15]

Advanced Examples

One prominent example of a multivariate concave function arises in information theory with the Shannon entropy, defined for a probability vector p = (p_1, \dots, p_n) in the simplex as H(p) = -\sum_{i=1}^n p_i \log p_i. This function is concave on the probability simplex, as established by the concavity of the negative entropy term and Jensen's inequality applied to the convex function x \log x.^[16] In production theory, the Cobb-Douglas function f(x, y) = x^a y^{1-a} for x, y \geq 0 and a \in (0,1) is concave on the nonnegative orthant, since its Hessian matrix is negative semidefinite, reflecting constant returns to scale and diminishing marginal returns.^[17] Log-linear utility functions, such as u(x) = \log\left( \sum_{i=1}^n x_i \right) for x \in \mathbb{R}^n_{++}, are concave because the summation is affine and the logarithm is a concave nondecreasing function, preserving concavity under composition. The extended-real-valued indicator function of a convex set C \subseteq \mathbb{R}^n, defined as I_C(x) = 0 if x \in C and -\infty otherwise, is concave, as its hypograph coincides with the convex set C \times \mathbb{R}. Composition rules further illustrate advanced concavity: if g is concave and nondecreasing on [0, \infty) and f is concave with nonnegative values, then g \circ f is concave; for instance, \sqrt{f(x)} inherits concavity when f is nonnegative and concave.

Applications

Optimization Theory

In optimization theory, concave functions play a pivotal role in maximization problems due to their desirable properties that ensure global optimality. For an unconstrained concave function f defined on a convex domain, any local maximum is also a global maximum, as the function's graph lies below any tangent line, preventing higher values elsewhere. This property simplifies the search for optima, as local optimization techniques suffice to identify the global solution.^[6] When maximizing a concave objective function subject to convex constraints, the problem can be reformulated as a convex minimization problem by negating the objective, allowing the application of standard convex optimization techniques. Specifically, maximizing concave f(x) over a convex set is equivalent to minimizing the convex function -f(x) over the same set. The Karush-Kuhn-Tucker (KKT) conditions, which include stationarity, primal and dual feasibility, and complementary slackness, are then necessary and sufficient for global optimality under a constraint qualification such as Slater's condition. Thus, any point satisfying the KKT conditions for this reformulated problem yields the global maximum of the original concave maximization.^[6] For solving these problems algorithmically, gradient ascent is commonly used for smooth concave functions, iteratively updating the solution in the direction of the positive gradient to approach the maximum. For nonsmooth concave functions, subgradient methods extend this approach by selecting a subgradient from the subdifferential at each step and performing ascent updates, converging to an optimal solution under appropriate step-size rules. If the function is strictly concave, the maximizer is unique, ensuring a single global optimum.^[6]^[18]^[19]

Economics and Utility Theory

In economics, concave utility functions are fundamental to modeling consumer behavior, as their concavity reflects the principle of diminishing marginal utility, where additional units of a good provide progressively smaller increments in satisfaction. This property ensures that the second derivative of the utility function with respect to consumption is negative, leading to risk-averse preferences under uncertainty. Specifically, for a twice-differentiable utility function u(x), concavity implies u''(x) < 0, which aligns with the Arrow-Pratt measure of absolute risk aversion defined as r_A(x) = -\frac{u''(x)}{u'(x)} > 0, quantifying how much an individual dislikes risk.^[20] This framework, originating from expected utility theory, explains why risk-averse agents prefer certain outcomes over gambles with the same expected value, as demonstrated by Jensen's inequality applied to concave utilities.^[21] Concave production functions similarly capture decreasing returns to scale in economic modeling, where output increases at a diminishing rate with additional inputs, promoting realistic depictions of resource constraints and efficiency. A prominent example is the Constant Elasticity of Substitution (CES) production function, given by Y = A \left[ \sum_i \alpha_i K_i^\rho + (1 - \sum_i \alpha_i) L^\rho \right]^{1/\rho} with \rho \leq 1, which exhibits concavity under these parameters and allows for flexible substitution between factors like capital (K) and labor (L) while maintaining decreasing marginal productivity. This form has been widely adopted in growth models to analyze technological progress and factor shares, as it balances substitutability with concavity to avoid unbounded outputs.^[22] In consumer theory, concave utility functions interacting with linear budget constraints yield convex indifference curves, ensuring unique tangency solutions for optimal consumption bundles and supporting the existence of demand functions. The upper contour sets of such utilities are convex, implying that indifference curves bow inward toward the origin, which facilitates the graphical representation of trade-offs and substitution effects under price changes. This convexity is a direct consequence of the quasi-concavity of the utility (strengthened to strict concavity for smoother behavior), enabling the second-order conditions for maximization to hold in standard models.^[21] Welfare economics employs concave social welfare functions to aggregate individual utilities in a manner that values equity, with the utilitarian variant—defined as W = \sum_i u_i(x_i)—being concave if each u_i is concave, thereby promoting Pareto-efficient allocations that weigh total welfare without excessive inequality tolerance. This approach underpins theorems like the First Welfare Theorem, where competitive equilibria maximize such functions under ideal conditions, and informs policy evaluations by penalizing distributions with high variance in utilities.^[21] An empirical application appears in portfolio theory, where the logarithmic utility function u(w) = \log(w) models risk-averse investors seeking to maximize long-term growth, as its relative risk aversion of unity leads to myopic allocation strategies independent of horizon length. In continuous-time settings, this yields the optimal fraction invested in the market portfolio as \pi = \frac{\mu - r}{\sigma^2}, balancing expected return \mu against volatility \sigma^2, and has been validated in asset allocation models for its simplicity and alignment with observed diversification behavior.^[23]

References

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Dec 6, 2022 · A function f is concave if for all x,y ∈ S and 0 ≤ t ≤ 1, f tx + (1 − t)y ≥ tf(x) + (1 − t)f(y). Every chord of the graph lies on or below the ...
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