The following article was written by ChatGPT, after some back-and-forth.

Abstract

In optimization and economic theory, the curvature of an objective function—whether concave or convex—profoundly influences resource allocation strategies. The Concave-Convex Allocation Principle posits that maximizing concave functions typically results in concentrated allocations of resources, whereas minimizing convex functions encourages diversified allocations. This principle is fundamental across various domains, guiding effective decision-making in resource distribution.

Introduction

Resource allocation is a central challenge in fields such as economics, finance, engineering, and operations research. The optimal distribution of limited resources depends significantly on the nature of the objective function. The Concave-Convex Allocation Principle provides a unifying framework to understand how the curvature of this function dictates whether resources should be concentrated or diversified.

Fundamentals of Concave and Convex Functions

Concave Function: A function $f: \mathbb{R}^n \to \mathbb{R}$ is concave if, for any $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ and $\theta \in [0, 1]$,

$$f(\theta \mathbf{x} + (1 - \theta) \mathbf{y}) \geq \theta f(\mathbf{x}) + (1 - \theta) f(\mathbf{y})$$

Intuition: The graph lies above the line segment connecting any two points.

Convex Function: A function $f: \mathbb{R}^n \to \mathbb{R}$ is convex if, for any $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ and $\theta \in [0, 1]$,

$$f(\theta \mathbf{x} + (1 - \theta) \mathbf{y}) \leq \theta f(\mathbf{x}) + (1 - \theta) f(\mathbf{y})$$

Intuition: The graph lies below the line segment connecting any two points.

Key Properties:

Concave: Any local maximum is a global maximum; Hessian is negative semi-definite.
Convex: Any local minimum is a global minimum; Hessian is positive semi-definite.

The Concave-Convex Allocation Principle

Principle Statement:

Maximizing concave functions under linear constraints leads to concentrated allocations, focusing resources on variables with the highest marginal returns.
Minimizing convex functions under linear constraints results in diversified allocations, spreading resources to balance marginal costs or risks.

Mathematical Formulation:

$$\begin{aligned} \max_{{a_i}} \quad & f(a_1, a_2, \ldots, a_n)\\ \text{subject to} \quad & \sum_{i=1}^n w_i a_i = c & a_i \geq 0, \quad \forall i \end{aligned}$$

Concave ( f ): Allocate all $c$ to the variable with the highest $\frac{\partial f}{\partial a_i} / w_i$, resulting in $a_j = \frac{c}{w_j}$ and $a_i = 0$ for $i \neq j$.
Convex ( f ): Distribute $c$ proportionally to weights, $a_i = \frac{c w_i^{-1}}{\sum_{j=1}^n w_j^{-1}}$, promoting diversification.

Illustrative Examples

Concave Maximization:

$$\max_{a_1, a_2} \quad f(a_1, a_2) = \sqrt{a_1} + \sqrt{a_2} \quad \text{subject to} \quad a_1 + a_2 = c$$

Optimal Allocation: $a_1 = c,\ a_2 = 0$ (or vice versa), demonstrating concentration.

Convex Minimization:

$$\min_{a_1, a_2} \quad f(a_1, a_2) = a_1^2 + a_2^2 \quad \text{subject to} \quad a_1 + a_2 = c$$

Optimal Allocation: $a_1 = a_2 = \frac{c}{2}$, illustrating diversification.

Investment Portfolio Optimization:

Concave Utility: Concentrate investments in the highest-return asset to maximize utility.
Convex Risk: Diversify investments to minimize overall portfolio risk.

Implications Across Domains

Economics: Firms concentrate resources on most profitable products under concave profit functions; diversify to minimize costs with convex functions.
Finance: Risk-averse investors concentrate holdings in secure assets; risk-minimizing strategies diversify portfolios.
Operations Management: Allocate resources to key processes for maximum efficiency or spread them to balance workloads and reduce costs.

Mathematical Insight: Jensen's Inequality

Jensen's Inequality formalizes the principle:

Concave ( f ):

$f\left( \sum_{i} \theta_i a_i \right) \geq \sum_{i} \theta_i f(a_i)$

Implication: Concentrating resources increases the objective value.

Convex ( f ):

$f\left( \sum_{i} \theta_i a_i \right) \leq \sum_{i} \theta_i f(a_i)$

Implication: Diversifying resources increases the objective value.

Conclusion

The Concave-Convex Allocation Principle provides a foundational guideline in optimization, linking the curvature of objective functions to resource distribution strategies. Recognizing whether an objective is concave or convex allows for informed decisions on whether to concentrate or diversify resources, enhancing effectiveness across economic, financial, and operational applications.

References:

Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic Theory. Oxford University Press.
Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91.

The Concave-Convex Allocation Principle, with ChatGPT