What is Optimization Problem

An optimization problem in mathematics is concerned with finding the best possible solution out of a set of feasible alternatives. The term "best" or "optimal" refers to a solution that maximizes or minimizes a certain function, often termed as the "objective function". Optimization problems are pervasive across many fields including operations research, computer science, machine learning, economics, and engineering.

Objective Function

At the heart of every optimization problem lies an objective function. This function represents a quantity that needs to be minimized or maximized. In the general form, an optimization problem can be defined as follows:

If we are seeking to maximize the function, the problem is defined as:

\max f(x)

If we are seeking to minimize the function, the problem is defined as:

\min f(x)

In both these cases, f(x) represents the objective function, and x is the decision variable or variables that we can control or change.

Constraints

Constraints are restrictions or conditions that the solution must satisfy. These could be equalities or inequalities involving the decision variables. The set of all points x that satisfy these constraints is called the feasible set. We denote the feasible set by S. Therefore, the general form of an optimization problem is:

If we are seeking to maximize the function:

\max f(x) \text{ subject to } x \in S

If we are seeking to minimize the function:

\min f(x) \text{ subject to } x \in S

In these equations, the constraints define the set S. The constraints could be represented in the form of equalities and inequalities as follows:

\begin{align*} g_i(x) & \leq 0, i = 1,...,m \\ h_j(x) & = 0, j = 1,...,p \end{align*}

In these equations, g_i(x) represents inequality constraints and h_j(x) represents equality constraints. The numbers m and p represent the number of inequality and equality constraints respectively.

Optimal Value and Optimal Solution

The optimal value of an optimization problem refers to the best possible value of the objective function given the constraints. If the problem is a minimization problem, the optimal value is the lowest possible value of the objective function over the feasible set. If it's a maximization problem, it's the highest possible value.

The optimal solution, denoted usually by x^*, is the particular value (or values) of the decision variable(s) x that yield the optimal value of the objective function. In mathematical terms:

If we are seeking to maximize the function, the optimal solution is defined as:

x^* = \arg\max_{x \in S} f(x)

If we are seeking to minimize the function, the optimal solution is defined as:

x^* = \arg\min_{x \in S} f(x)

In these equations, x^* represents the decision variable(s) that yield the optimal value of the objective function. Note that there may be more than one optimal solution in an optimization problem, i.e., more than one x^* that yield the optimal value of f(x).

Common Optimization Problems

In practice, optimization problems take various forms, depending on the nature of the objective function and the constraints, as well as the type of values the decision variables can take.

Continuous Optimization

Continuous optimization problems involve decision variables that can take any value within a specific continuous range. These problems are common in various areas like machine learning, scheduling, resource allocation, where the variables can assume a continuous range of values.

There are several types of continuous optimization problems, each characterized by the nature of the objective function and the constraints. For example, if the objective function and the constraints are both linear, the problem is a linear programming problem. However, if they are nonlinear, the problem is a nonlinear programming problem.

Linear Programming

Linear Programming (LP) is a fundamental type of continuous optimization problem where both the objective function and the constraints are linear. In a linear programming problem, each variable in the objective function and the constraints appears with a degree of one and is not multiplied by any other variable. LP problems are prevalent in areas like operations research, logistics, supply chain management, and more, due to their relative simplicity and the existence of efficient solution algorithms.

Nonlinear Programming

Nonlinear Programming (NLP) problems are optimization problems where the objective function, constraints, or both are nonlinear. Nonlinear programming allows for a much wider range of real-world problems to be modeled compared to linear programming. However, nonlinearity also introduces additional complexity into the problem.

Quadratic Programming

Quadratic programming is a specific type of nonlinear programming where the objective function is quadratic, and the constraints are linear. Quadratic programming is commonly used in machine learning and statistics for problems like support vector machines and portfolio optimization.

Quadratically Constrained Programming

Quadratically Constrained Programming (QCP) is a more complex form of quadratic programming where the constraints can also be quadratic. QCP problems can model a wider range of scenarios than simple quadratic programming but are also more challenging to solve.

Second-Order Cone Programming

Second-Order Cone Programming (SOCP) is a type of convex optimization problem that generalizes linear and quadratic programming. SOCP can handle a broader range of problems compared to LP and QP while still maintaining tractability and efficiency of optimization.

Semi-Definite Programming

Semi-Definite Programming (SDP) is another type of convex optimization where the decision variables are matrices, and the constraints require these matrices to be semi-definite. SDP finds applications in control theory, signal processing, and quantum information theory, among other fields.

Convex Programming

Convex Programming is a subclass of nonlinear programming where the objective function is convex, and the constraints form a convex set. Convex programming problems have the property that local minima are also global minima, which makes them particularly tractable among nonlinear programming problems.

Discrete Optimization

Discrete Optimization involves problems where the decision variables can only take on discrete values. Such problems often arise in fields such as computer science, logistics, operations research, and transportation, where decisions often involve a choice among a finite or countably infinite set of alternatives.

Integer Programming

Integer Programming (IP) problems involve decision variables that can only take integer values. These problems arise in scenarios where fractional values do not make sense, such as when determining the number of units to produce or the number of trucks required for delivery.

Mixed-Integer Programming

In Mixed-Integer Programming (MIP) problems, some decision variables are required to have integer values, while others can be continuous. These problems are common in fields like production planning and logistics, where decisions may involve a mix of discrete choices and continuous quantities.

Combinatorial Optimization

Combinatorial Optimization problems involve finding the best object from a finite set of objects. In these problems, the decision variables often represent whether or not to include a particular object in a solution, leading to a combinatorial number of potential solutions. Examples include the traveling salesman problem and the knapsack problem.

Global Optimization

Global Optimization problems involve finding the absolute optimal solution among all possible solutions, as opposed to finding a local optimum that cannot be improved upon by considering only neighboring solutions. These problems can be particularly challenging to solve, especially when the search space is large or the objective function has many local optima. Global Optimization techniques often employ strategies to escape from local optima in search of the global optimum.

Stochastic Programming

Stochastic Programming is a framework for modeling optimization problems that involve uncertainty. Whereas traditional optimization deals with finding the best decision given a certain set of inputs and constraints, stochastic programming introduces random variables into the model to represent the uncertainty. The resulting optimization problem aims to find the best decision given the probability distributions of the uncertain parameters.