Equality

Last Updated: 05/09/2024

Solving optimization problems with quadratic equality constraints

One thing I find odd about optimization theory is the division between the objective and the constraints. In a way, separating the two is a workaround for the limitations of dealing with infinity. To see what I mean, consider the following unconstrained optimization problem:

\[\argmin_{x, y}\: (x-a)^2 + (y-b)^2 + \lambda(x-y)^2\]

What is the effect of the $\lambda (x-y)^2$ term? For $\lambda=0$, the solution is simply $x=a$ and $y=b$, setting the objective to 0. But as $\lambda$ increases it forces $x$ and $y$ to be closer and closer. In the limit $\lambda\rightarrow\infty$, it is easy to see $x=y$. Mathematically, the problem starts to resemble the constrained optimization problem:

\[\argmin_{x, y}\: (x-a)^2 + (y-b)^2 \\ \mathrm{s.t.}\:\:x = y\]

In fact, the unconstrained problem has a closed form solution for any choice of constants $a$, $b$, $\lambda$. The solution reflects this limiting behavior:

\[x = \frac{(1+\lambda)a + \lambda b}{2\lambda+1} \Rightarrow x = \frac{a/\lambda + a + b}{2+1/\lambda}\]

In the limit as $\lambda\rightarrow\infty$, $x = y = (a+b)/2$. It is now clear that the $\lambda(x-y)^2$ term performs exactly the same function as the constraint $x=y$ for $\lambda\rightarrow\infty$.

What’s going on here? Can we move terms between the objective and constraint set through weights in general? The nonconvexity of problems with convex quadratic equality constraints suggests the answer is no, that limit arguments can only take you so far. To start an answer it helps to consider places where this objective trick is almost used: relaxations and dual problems, which often move the constraints into the objective.

Relaxations and Duals

Generalizing

\(\argmin_{\mathbf{x}\in\mathbb{R}^n}\: \mathbf{f}(\mathbf{x}) + \lambda \mathbf{g}(\mathbf{x})\)

To make this explicit, consider the canonical optimization problem:

\[\argmin_{x\in\mathbb{R}^n}\: f_0(x) \\ \mathrm{s.t.}\:\:\: f_i(x) \leq 0\\ \quad\:\:\:\: h_j(x) = 0\]

Suspending careful consideration of infinity, we could have written:

\[\argmin_{x\in\mathbb{R}^n}\: f_0(x) + \sum_i\infty\cdot f_i(x) + \sum_j\infty\cdot\|h_j(x)\|^2\]

Allow, for a moment, $\infty\cdot0 = 0$. Then, this new objective is infinity if any of the inequality or equality constraints are broken. If $h_j(x) != 0$, the entire objective goes to inifinty.

Clearly, there are a couple of problems with this. Besides using $\infty$ as a coefficient, the inequality constraints $f_i(x)$ don’t play as nicely. If $f_i(x)<0$ the objective becomes negative infinity, and the other terms cease to matter. But hopefully the basic intuition is clear, at least for the equality constraints.

Motivating

This fluidity between constraints and objective terms seems to an implicit part of how we solve problems. Lagrangian duality

An example

Consider the two dimensional problem: \(\min_{x,y}\:(x-2y)^2 \quad\mathrm{s.t.}\:\; x^2 + y^2 = 1\)