complementarity_constraints.qmd

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title: "Complementarity constraints"
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---

## Why complementarity constraints?

In this chapter we are going to present yet another framework for modelling hybrid systems, which comes with a rich theory and efficient algorithms. It is based on *complementarity constraints*. Before we introduce the modelling framework in the next section, we first explain the very concept of complementarity constraints and the related optimization problems.

## Definition of complementarity constraints

Two variables $x\in\mathbb R$ and $y\in\mathbb R$ satisfy the c*omplementarity constraint* if $x$ or $y$ is equal to zero and both are nonnegative 

$$xy=0, \; x\geq 0,\; y\geq 0,$$  

or, using a dedicated compact notation  
$$\boxed{0\leq x \perp y \geq 0.}$$

::: {.callout-warning}
## Both variables can be zero
The *or* in the above definition is not exclusive, therefore it is possible that both $x$ and $y$ are zero.
:::

The concept and notation extends to vectors $x\in\mathbb R^n$ and $y\in\mathbb R^n$, in which case the constraint is interpreted componentwise
$$\boxed{\bm 0\leq \bm x \perp \bm y \geq \bm 0.}$$

## Geometric interpretation of complementarity constraints 

The set of admissible pairs $(x,y)$ in the $\mathbb R^2$ plane is constrained to the L-shaped subset given by the nonnegative $x$ and $y$ semi-axes (including the origin). 

![The set of solutions satisfying a complementarity constraint](complementarity_figures/l_shaped_complementarity_set.png){width=30%}

Optimization over these constraints is difficult, and not only because the feasible set is nonconvex, but also because *constraint qualification* conditions are not satisfied. Still, some results and tools are available for some classes of optimization problems with these constraints.

## Linear complementarity problem (LCP) {.scrollable}

For a given square matrix $\mathbf M$ and a vector $\mathbf q$ , the *linear complementarity problem* (LCP) asks for finding two vectors $\bm w$ and $\bm z$ satisfying
$$
  \begin{aligned}
  \bm w-\mathbf M\bm z &= \mathbf q \\
  \bm 0 \leq \bm w &\perp \bm z \geq \bm 0.
  \end{aligned}
$$

Just by moving all the provided data to the right hand side we get
$$
  \begin{aligned}
  \bm w &= \underbrace{\mathbf M\bm z + \mathbf q}_{\mathbf f(\bm z)} \\
  \mathbf 0 \leq \mathbf f(\bm z) &\perp \bm z \geq \mathbf 0,
  \end{aligned}
$$  
from which we can immediately guess how the linear problem needs to be modified so that we get a *nonlinear complementarity problem* (NLCP).  

## Existence of a unique solution 

A unique solution exists for every vector $\mathbf q$ if and only if the matrix $\mathbf M$ is a P-matrix (something like positive definite, but not exactly, look it up yourself).

## LCP related to LP and QP {.scrollable}

Note that the KKT conditions for LP and QP come in the form of LCP.

Consider the QP problem with inequality constraints
$$\text{minimize} \quad \frac{1}{2}\bm x^\top \mathbf Q \bm x + \mathbf c^\top \bm x$$ $$\text{subject to} \quad \mathbf A\bm x \geq \mathbf b,\quad \bm x\geq \mathbf 0,$$	

The KKT conditions can be written (and compare these to the conditions for the equality-constrained QP)
$$
\begin{aligned}
\mathbf 0\leq \bm x &\perp \mathbf Q\bm x - \mathbf A^\top \bm \lambda + \mathbf c \geq \mathbf 0\\
\mathbf 0\leq \bm \lambda &\perp \mathbf A\bm x -\mathbf b \geq \mathbf 0.
\end{aligned}
$$	

These can be reformatted as
$$
\mathbf 0\leq \underbrace{\begin{bmatrix}\bm x\\ \bm \lambda\end{bmatrix}}_{\bm z} 
\perp
\underbrace{\begin{bmatrix} \mathbf Q & -\mathbf A^\top \\ \mathbf A & \mathbf 0\end{bmatrix}\begin{bmatrix}\bm x\\ \bm \lambda\end{bmatrix} + \begin{bmatrix} \mathbf c \\ -\mathbf b \end{bmatrix}}_{\mathbf M\bm z+\mathbf q}\geq 0.$$ 

## Nonlinear complementarity problem

Given a vector function $\mathbf f: \mathbb R^n\rightarrow \mathbb R^n$, find a vector $\bm x\in\mathbb R^n$ satisfying
$$
\bm 0\leq \bm x \perp \mathbf f(\bm x) \geq \bm 0.
$$

## Mixed complementarity problem (MCP) {.scrollable}

An extension of complementarity constraint to the situation in which the variable $x$ is lower- and upper-bounded. In particular, it can be stated as
$$
  l \leq x \leq u \perp f(x).
$$  

The convention for interpretation is

- If $x$ is strictly within the interval, that is, $l < x < u$ , then $f(x)=0$,
- If $x=l$ , then $f(x)\geq 0$ ,
- if $x=u$ , then $f(x)\leq 0$ .

## Extended linear complementarity problem (ELCP) {.scrollable}

Given some matrices $\mathbf A$ and $\mathbf B$ , vectors $\mathbf c$ and $\mathbf d$ , and $m$ subsets $\phi_j \sub \{1,2,\ldots,p\}$ , find a vector $\bm x$ such that
$$
  \begin{aligned}
  \sum_{j=1}^m\prod_{i\in\phi_j}(\mathbf A\bm x - \mathbf c)_i &= 0,\\
  \mathbf A\bm x &\geq \mathbf c,\\
  \mathbf B\bm x &= \mathbf d,
  \end{aligned}
$$  
or show that no such $\bm x$ exists.

The first equation is equivalent to 
$$
\forall j \in \{1, \ldots, m\} \; \exist i \in \phi_j \;\text{such that} \; (\mathbf A\bm x − \mathbf c)_i = 0.
$$

Geometric interpretation: union of some faces of a polyhedron.

## Mathematical program with complementarity constraints (MPCC)

The mathematical program with complementarity constraints (MPCC) is 
$$
  \begin{aligned}
  \operatorname*{minimize}_{\bm x\in\mathbb R^n} & \;f(\bm x)\\
  \text{subject to} & \;0\leq h(\bm x) \perp g(\bm x) \geq 0.
  \end{aligned}
$$

Special case of *Mathematical program with equilibrium constraints (MPEC)*.

## Mathematical program with equilibrium constraints (MPEC)

Optimization problem in which some variable should satisfy equilibrium constraints:
$$
  \begin{aligned}
  \min_{x_1,x_2} &\; f(x_1,x_2)\\
  \text{subject to}&\; \nabla_{x_2} \phi(x_1,x_2) = 0
  \end{aligned}
$$

For convex $\phi()$ it can be reformulated into a *Bilevel optimization* problem.

## Bilevel optimization

Optimization problem in which some variables are constrained to be results of some inner optimization.
In the simplest form
$$
\begin{aligned}
  \min_{x_1,x_2} &\; f(x_1,x_2)\\
  \text{s. t.}\ &\; x_2 = \text{arg}\,\min_{x_2} \;\phi(x_1,x_2)
\end{aligned}
$$

## Disjunctive constraints

A number of affine constraints combined with $\lor$ and $\land$ logical operators.

$$
T_1 \lor T_2 \lor \ldots \lor T_m,
$$
where 
$$
T_i = T_{i1} \land T_{i1} \land \ldots \land T_{in_{i}},
$$ 
where 
$$
T_{ij} = c_{ij}x + d_{ij} \in \mathcal D_{ij}.
$$