Fundamentals of Optimization

Definition 23

The standard form of optimization is

$p^\star = \min_\mathbf{x} f_0(\mathbf{x}) \text{ such that } f_i(\mathbf{x}) \leq 0$

• The vector $\mathbf{x}\in\mathbb{R}^n$ is known as the decision variable.

• The function $f_0:\mathbb{R}^n\to\mathbb{R}$ is the objective.

• The functions $f_i:\mathbb{R}^n\to\mathbb{R}$ is the constraints.

• $p^\star$ is the optimal value, and the $\mathbf{x}^\star$ which achieves the optimal value is called the optimizer.

Definition 24

The feasible set of an optimization problem is

$\mathcal{X} = \{\mathbf{x}\in\mathbb{R}^n:\ f_i(\mathbf{x}) \leq 0 \}$

Definition 25

A point $\mathbf{x}$ is $\epsilon$-suboptimal if it is feasible and satisfies

$p^\star \leq f_0(\mathbf{x}) \leq p^\star + \epsilon$

Definition 26

An optimization problem is strictly feasible if $\exists \mathbf{x}_0$such that all constraints are strictly satisfied (i.e inequalities are strict inequalities, and equalities are satisfied).

Problem Transformations

Sometimes, optimizations in a particular formulation do not admit themselves to be solved easily. In this case, we can sometimes transform the problem into an easier one from which we can easily recover the solution to our original problem. In many cases, we can introduce additional “slack” variable and constraints to massage the problem into a form which is easier to analyze.

Theorem 7 (Epigraphic Constraints)

$\min_\mathbf{x} f_0(x)$ is equivalent to the problem with epigraphic constraints

$\min_{\mathbf{x}, t} t \quad : \quad f_0(x) \leq t,$

Theorem 7 works because by minimizing $t$, we are also minimizing how large $f_0(x)$ can get since $f_0(x) \leq t$, so at optimum, $f_0(x) = t$. It can be helpful when $f_0(x) \leq t$ can be massaged further into constraints that are easier to deal with.

Theorem 8 (Monotone Objective Transformation)

Let $\Phi:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly increasing function over a feasible set $\mathcal{X}$. Then

$\min_{\mathbf{x}\in\mathcal{X}}f_0(\mathbf{x}) \equiv \min_{\mathbf{x}\in\mathcal{X}} \Phi(f_0(\mathbf{x}))$

Robust Optimization

For a “nominal” problem

$\min_\mathbf{x} f_0(\mathbf{x}) \quad : \quad \forall i\in[1,m],\ f_i(\mathbf{x}) \leq 0,$

uncertainty can enter in the data used to create the $f_0$ and $f_i$. It can also enter during decision time where the $\mathbf{x}^\star$ which solves the optimization cannot be implemented exactly. These uncertainties can create unstable solutions or degraded performance. To make our optimization more robust to uncertainty, we add a new variable $\mathbf{u}\in\mathcal{U}$.

Definition 27

For a nominal optimization problem $\min_\mathbf{x} f_0(\mathbf{x})$ subject to $f_i(\mathbf{x}) \leq 0$ for $i\in[1,m]$, the robust counterpart is

$\min_\mathbf{x} \max_{\mathbf{u}\in\mathcal{U}} f_0(\mathbf{x}, \mathbf{u}) \quad : \quad \forall i\in[1,m],\ f_i(\mathbf{x}, \mathbf{u}) \leq 0$