A feasible solution is a candidate assignment that satisfies every constraint in a constrained optimization problem. Optimization searches only feasible points, choosing the one with the highest objective value; a point that violates any constraint is infeasible. section 8.1.1 section 8.6
A constraint can exclude the unconstrained optimum
Suppose a one-dimensional objective is maximized subject to a cost constraint c ≤ 0. The region with x > 4 contains the unconstrained global optimum, but it is infeasible. The best feasible solution lies in the shaded allowed regions and achieves an objective value of roughly 4 instead of more than 8. section 8.1.1
In constrained optimization, feasibility guides the search:
- Reject or rule out points that violate the predefined constraints. section 8.1.1
- Compare objective values only among points satisfying the constraints. section 8.6
- In constrained Bayesian optimization, estimate the probability of feasibility with Gaussian processes and incorporate it into expected improvement. section 8.6
