Epsilon-Constrained Optimization

[Update 2022-10-18: Replace epsilon with delta in description of example problem 7 in pgapack]

Many optimization problems involve constraints that valid solutions must satisfy. A constrained optimization problem is typically written as a nonlinear programming problem [1].

\begin{align*} \hbox{Minimize} \; & f_i(\vec{x}), & i &= 1, \ldots, I \\ \hbox{Subject to} \; & g_j(\vec{x}) \le 0, & j &= 1, \ldots, J \\ & h_k(\vec{x}) = 0, & k &= 1, \ldots, K \\ & x_m^l \le x_m \le x_m^u, & m &= 1, \ldots, M \\ \end{align*}

In this problem, there are \(n\) variables (the vector \(\vec{x}\) is of size \(n\)), \(J\) inequality constraints, \(K\) equality constraints and the variable \(x_m\) must be in the range \(|x_m^l, x_m^u|\) (often called a box constraint). The functions \(f_i\) are called the objective functions. If there is more than one objective function, the problem is said to be a multi objective optimization problem as described previously in this blog [2]. In the following I will use some terms that were introduced in that earlier blog-post without further explanation.

The objective functions are not necessarily minimized (as in the formula) but can also be maximized if the best solutions to a problem requires maximization. Note that the inequality constraints are often depicted with a \(\ge\) sign but the formula can easily be changed (e.g. by multiplying with -1) to use a \(\le\) sign.

Since it is very hard to fulfil equality constraints, especially if they involve nonlinear functions of the input variables, equality constraints are often converted to inequality constraints using an δ‑neighborhood:

\begin{equation*} -\delta \le h_k(\vec{x}) \le \delta \end{equation*}

Where δ is chosen according to the requirements of the problem for which a solution is sought.

One very successful method of solving constrained optimization problems is to consider a lexicographic ordering of constraints and objective function values. Candidate solutions to the optimization problem are first sorted by violated constraints (typically the sum over all violated constraints) and then by the objective function value(s) [1]. When comparing two individuals during selection in the genetic algorithm there are three cases: If both individuals violate constraints, the individual with the lower constraint violation wins. If one violates constraints and the other does not, the one not violating constraints wins. In the last case where both individuals do not violate constraints the normal comparison is used (which depends on the algorithm and if we're minimizing or maximizing). This method, originally proposed by Deb [1], is implemented in the genetic algorithm package I'm maintaining, PGAPack, and my Python wrapper PGAPy for it.

With this algorithm for handling constraints, the constraints are optimized first before the algorithm "looks" at the objective function(s) at all. It often happens that the algorithm ends up searching in a region of the input space where no good solutions exist (but no constraints are violated). Hard problems often contain equality constraints (converted to inequality constraints as indicated earlier) or other "hard" constraints. In my previous blog post [2] on antenna optimization I wrote: "the optimization algorithm has a hard time finding the director solutions at all. Only in one of a handful experiments I was able to obtain the pareto front plotted above".

In that experiment I was running 50 searches and only 5 of them did not get stuck in a local optimum. A similar thing happens for the problem (7) in Deb's paper [1] which has equality constraints. I've implemented this as example problem 7 in PGAPack. It only finds a solution near the (known) optimum when \(\delta \ge 10^{-2}\) for all equality constraints (I didn't experiment with different random seeds for the optimizer, maybe a better solution would be possible with a different random seed). In the paper [1], Deb uses \(\delta = 10^{-3}\) for the same reason.

One method for handling this problem was appealing because it is so easy to understand and implement: Takahama and Sakai were first experimenting with a method for relaxing constraints during the early stages of optization with a formulation they called an α‑constrained genetic algorithm [3]. They later simplified the formulation and called the resulting algorithm ε constrained optimization. It can be applied to different optimization algorithms, not just genetic algorithms and variants [4]. Of special interest is the application of the method to differential evolution [5], [6] but of course it can also be applied to other forms of genetic algorithms.

Note that the ε in the name of the algorithm can be used for the δ used when converting an equality constraint to inequality constraints but is not limited to this case.

During the run of the optimizer in each generation a new value for ε is computed. The comparison of individuals outlined above is modified, so that an individual is handled like it was not violating any constraints if the constraint violation is below ε. So if both individuals have constraint violations larger than ε, the one with lower violation wins. If one violation is below ε and the other above, the individual with the violation below ε wins. And finally if the constraint violations of both individuals are below ε, the normal comparison takes place.

The last case is the key to the success of this algorithm: Even though the search proceeds into a direction where the constraint violations are minimized, at the same time good solutions in terms of the objective function are found.

The algorithm begins by initializing \(\varepsilon_0\) with the constraint violation of the individuum with index \(\theta\) from the initial population sorted by constraint violation, where \(\theta\) is a parameter of the algorithm between 1 and the population size, a good value uses the individuum at about 20% of the population size which is also the default in PGAPack. In each generation \(t\), \(\varepsilon_t\) is computed by

\begin{equation*} \varepsilon_t = \varepsilon_0 \left(1-\frac{t}{T_c}\right)^{cp} \end{equation*}

up to generation \(T_c\). After that generation, ε is set to 0. The exponent \(cp\) is between 2 and 10. The 2010 paper [6] recommends to set \(cp = 0.3 cp + 0.7 cp_\min\) at generation \(T_\lambda = 0.95 T_c\) where \(cp_\min\) is the fixed value 3. The initial value of \(cp\) is chosen so that \(\varepsilon_\lambda=10^{-5}\) at generation \(T_\lambda\) unless it is smaller than \(cp_\min\) in which case it is set to \(cp_\min\). PGAPack implements this schedule for \(cp\) by default but allows to change \(cp\) at start and during run of the optimizer, so it's possible to easily implement a different schedule for \(cp\) – the default works quite nicely, though.

With the ε constraint method, example 7 from Deb [1] can be optimized with a precision of \(10^{-6}\) in my experiments, see the epsilon_generation parameter in the optimizer example

The antenna-optimizer with an ε‑generation of 50 (that's the \(T_c\) parameter of the algorithm) gets stuck in the local optimum only in one of 50 cases, all other cases find good results:

In that picture all the solutions that are dominated by solutions from another run are drawn in black. It can be seen that the data from run number 16 did not contribute any non-dominated solutions (on the right side in the legend the number 16 is missing). You can turn off the display of the dominated solutions by clicking on the black dot in the legend.

When I increase the ε‑generation to 60, the run with random seed 16 also finds a solution:

We also see that the solutions are quite good (quite near to the pareto front) for all runs, the black "shadow" of the dominated solutions is quite near to the real pareto front and it is enough to do a single run of the algorithm for finding a good set of solutions.