Fundamentals of Optimization

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Introduction
Linear Algebraic Optimization
Fundamentals of Optimization
Definition 23
The standard form of optimization is
​p⋆=min⁡xf0(x) such that fi(x)≤0p^\star = \min_\mathbf{x} f_0(\mathbf{x}) \text{ such that } f_i(\mathbf{x}) \leq 0p⋆=minx​f0​(x) such that fi​(x)≤0
​
The vector x∈Rn\mathbf{x}\in\mathbb{R}^nx∈Rn
 is known as the decision variable.
The function f0:Rn→Rf_0:\mathbb{R}^n\to\mathbb{R}f0​:Rn→R
 is the objective.
The functions fi:Rn→Rf_i:\mathbb{R}^n\to\mathbb{R}fi​:Rn→R
 is the constraints.
​p⋆p^\starp⋆
 is the optimal value, and the x⋆\mathbf{x}^\starx⋆
 which achieves the optimal value is called the optimizer.
Definition 24
The feasible set of an optimization problem is
​X={x∈Rn: fi(x)≤0}\mathcal{X} = \{\mathbf{x}\in\mathbb{R}^n:\ f_i(\mathbf{x}) \leq 0 \}X={x∈Rn: fi​(x)≤0}
​
Definition 25
A point x\mathbf{x}x
 is ϵ\epsilonϵ
-suboptimal if it is feasible and satisfies
​p⋆≤f0(x)≤p⋆+ϵp^\star \leq f_0(\mathbf{x}) \leq p^\star + \epsilonp⋆≤f0​(x)≤p⋆+ϵ
​
Definition 26
An optimization problem is strictly feasible if ∃x0\exists \mathbf{x}_0∃x0​
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⁡xf0(x)\min_\mathbf{x} f_0(x)minx​f0​(x)
 is equivalent to the problem with epigraphic constraints
​min⁡x,tt:f0(x)≤t,\min_{\mathbf{x}, t} t \quad : \quad f_0(x) \leq t,minx,t​t:f0​(x)≤t,
​
Theorem 7 works because by minimizing ttt
, we are also minimizing how large f0(x)f_0(x)f0​(x)
 can get since f0(x)≤tf_0(x) \leq tf0​(x)≤t
, so at optimum, f0(x)=tf_0(x) = tf0​(x)=t
. It can be helpful when f0(x)≤tf_0(x) \leq tf0​(x)≤t
 can be massaged further into constraints that are easier to deal with.
Theorem 8 (Monotone Objective Transformation)
Let Φ:R→R\Phi:\mathbb{R}\to\mathbb{R}Φ:R→R
 be a continuous and strictly increasing function over a feasible set X\mathcal{X}X
. Then
​min⁡x∈Xf0(x)≡min⁡x∈XΦ(f0(x))\min_{\mathbf{x}\in\mathcal{X}}f_0(\mathbf{x}) \equiv \min_{\mathbf{x}\in\mathcal{X}} \Phi(f_0(\mathbf{x}))minx∈X​f0​(x)≡minx∈X​Φ(f0​(x))
​
Robust Optimization
For a “nominal” problem
​min⁡xf0(x):∀i∈[1,m], fi(x)≤0,\min_\mathbf{x} f_0(\mathbf{x}) \quad : \quad \forall i\in[1,m],\ f_i(\mathbf{x}) \leq 0,minx​f0​(x):∀i∈[1,m], fi​(x)≤0,
​
uncertainty can enter in the data used to create the f0f_0f0​
 and fif_ifi​
. It can also enter during decision time where the x⋆\mathbf{x}^\starx⋆
 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 u∈U\mathbf{u}\in\mathcal{U}u∈U
.
Definition 27
For a nominal optimization problem min⁡xf0(x)\min_\mathbf{x} f_0(\mathbf{x})minx​f0​(x)
 subject to fi(x)≤0f_i(\mathbf{x}) \leq 0fi​(x)≤0
 for i∈[1,m]i\in[1,m]i∈[1,m]
, the robust counterpart is
​min⁡xmax⁡u∈Uf0(x,u):∀i∈[1,m], fi(x,u)≤0\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 0minx​maxu∈U​f0​(x,u):∀i∈[1,m], fi​(x,u)≤0
​
Linear Algebra
Linear Algebraic Optimization
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Contents
Problem Transformations
Robust Optimization