Mathematical Optimization

Identifying the optimal solutions to mathematically defined problems.

What is mathematical optimization?

Mathematical Optimization (a.k.a. Nonlinear Programming, Mathematical Programming, or Numerical Optimization) encompasses the discipline of identifying the optimal solutions to mathematically defined problems (Snyman et al. 2005).

Formally, Mathematical Optimization is the process of (1) the formulation and (2) the solution of a constrained optimization problem of the general mathematical form:

$$\underset{\mathbf{x}}{min}f(\mathbf{x}),\mathbf{x}=[x_1,x_2,...x_n{]}^{\mathrm{\top }}\in {\mathbb{R}}^n$$

subject to the constraints

$$\begin{array}{}\text{(7.1)}& \begin{array}{c}{g}_j(\mathbf{x})\le 0,j=1,2,...,m\hfill \\ \hfill h_j(\mathbf{x})=0,j=1,2,...,r\end{array}\end{array}$$

where $f(\mathbf{x})$, $g_j(\mathbf{x})$, and $h_j(\mathbf{x})$ are scalar functions of the real column vector $\mathbf{x}$. The continuous components $x_i$ of $\mathbf{x}=[x_1,x_2,...x_n{]}^{\mathrm{\top }}$ are called the (design) variables, $f(\mathbf{x})$ is the objective function, $g_j(\mathbf{x})$ denotes the respective inequality constraint functions and $h_j(\mathbf{x})$ the equality constraint functions.

The optimum vector $\mathbf{x}$ that solves problem stated in equation 7.1 is denoted by ${\mathbf{x}}^{\ast }$ with corresponding optimum function value $f({\mathbf{x}}^{\ast })$. If no constraints are specified, the problem is called an unconstrained optimization problem.

Snyman, J. A., D. N. Wilke, and others. 2005. Practical mathematical optimization. (P. M. Pardalos and D.-Z. Du, Eds.). Second edition. Springer.