Foundations of Mathematical Optimization

- Overview

- Optimization Problems

In mathematical optimization, a problem is defined by an objective function, constraints, and decision variables. The objective function is a mathematical expression that quantifies the goal to be achieved, either maximized or minimized. Constraints are limitations or restrictions that define the feasible region, limiting the possible values of the decision variables. Decision variables are the unknown parameters that can be adjusted to influence the outcome of the problem.  

• Objective Function: This function represents the goal of the optimization problem. It is a mathematical expression that maps the decision variables to a scalar value, which is then either minimized or maximized. For example, in a cost minimization problem, the objective function would be the total cost, and the goal would be to find the values of the decision variables that minimize this cost. 
• Constraints: Constraints are inequalities or equalities that restrict the feasible solution space. They define the boundaries within which the decision variables can take values. For example, in a production planning problem, constraints might include resource limitations, production capacity, or demand requirements. 
• Decision Variables: These are the unknown parameters or variables that can be adjusted to influence the outcome of the optimization problem. Finding the optimal values of the decision variables that satisfy the constraints and optimize the objective function is the main goal of the optimization process. 
• Algorithms: Various algorithms are employed to solve optimization problems, depending on the problem's characteristics and complexity. These include mathematical programming methods like linear programming, nonlinear programming, and heuristic algorithms like genetic algorithms, simulated annealing, and particle swarm optimization. 
• High-Dimensional Spaces: In many real-world problems, the number of decision variables can be very large, leading to a high-dimensional solution space. This can make it computationally challenging to find the optimal solution using traditional optimization algorithms. Specialized algorithms and techniques are often needed to effectively explore and exploit these high-dimensional spaces.