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Mathematical Optimization

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.1)

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding “best available” values of some objective function given a defined domain (or a set of constraints), including a variety of different types of objective functions and different types of domains..

Optimization Problems

An optimization problem can be represented in the following way:

Given: a function $f : A \to R$ from some set $A$ to the real numbers
Sought: an element $x_0$ in $A$ such that $f(x_0) \leq f(x)$ for all $x$ in $A$ (“minimization”) or such that $f(x_0) \geq f(x)$ for all $x$ in $A$ (“maximization”).

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physics and computer vision may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled.

Typically, $A$ is some subset of the Euclidean space $\mathbb{R}^n$, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain $A$ of $f$ is called the search space or the choice set, while the elements of $A$ are called candidate solutions or feasible solutions.

By convention, the standard form of an optimization problem is stated in terms of minimization. Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima, where a local minimum x* is defined as a point for which there exists some $\delta > 0$ so that for all $x$ such that $$ \|\mathbf{x}-\mathbf{x}^*\|\leq\delta;\, $$ the expression $$f(\mathbf{x}^*)\leq f(\mathbf{x})$$ holds; that is to say, on some region around $x^*$ all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.


Optimization problems are often expressed with special notation. Here are some examples.

Minimum and maximum value of a function

Consider the following notation:

$$\min_{x\in\mathbb R}\; (x^2 + 1)$$ This denotes the minimum value of the objective function $x^2 + 1$, when choosing $x$ from the set of real numbers $\mathbb R$. The minimum value in this case is 1, occurring at $x = 0$.

Similarly, the notation

$$\max_{x\in\mathbb R}\; 2x$$ asks for the maximum value of the objective function $2x$, where $x$ may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is “infinity” or “undefined”.

Optimal input arguments

Consider the following notation: $$ \underset{x\in(-\infty,-1]}{\operatorname{arg\,min}} \; x^2 + 1, $$ or equivalently $$ \underset{x}{\operatorname{arg\,min}} \; x^2 + 1, \; \text{subject to:} \; x\in(-\infty,-1]. $$

This represents the value (or values) of the argument $x$ in the interval $(-\infty,-1]$ that minimizes (or minimize) the objective function $x^2 + 1$ (the actual minimum value of that function is not what the problem asks for). In this case, the answer is $x = -1$, since $x = 0$ is infeasible, i.e. does not belong to the feasible set.

Similarly, $$ \underset{x\in[-5,5], \; y\in\mathbb R}{\operatorname{arg\,max}} \; x\cos(y), $$ or equivalently $$ \underset{x, \; y}{\operatorname{arg\,max}} \; x\cos(y), \; \text{subject to:} \; x\in[-5,5], \; y\in\mathbb R, $$ represents the $(x,y)$ pair (or pairs) that maximizes (or maximize) the value of the objective function $x\cos(y)$, with the added constraint that $x$ lie in the interval $[-5,5]$ (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form $(5, 2k\pi)$ and $(-5,(2k+1)\pi)$, where $k$ ranges over all integers.

Arg min and arg max are sometimes also written argmin and argmax, and stand for argument of the minimum and argument of the maximum.

1) The Nature of Mathematical Programming, Mathematical Programming Glossary, INFORMS Computing Society.
mathematical_optimization.txt · Last modified: 1998/12/03 12:11 (external edit)