3.4 Local and Global Optima
Best nearby versus best anywhere¶
A feasible point is a local minimum if no sufficiently small feasible perturbation improves the objective. It is a global minimum if no feasible point anywhere has a lower objective:
Most realistic CCD problems are nonlinear and nonconvex. Natural frequencies move with geometry, controller gains change closed-loop poles, saturation introduces piecewise behavior, and architecture decisions create separated regions. A numerical solver will often provide a locally optimal design, not a proof of global optimality.
Why different initial designs matter¶
A low-stiffness plant with a large actuator may lie in a different basin of attraction from a stiff plant with modest control authority. Both can satisfy the same requirements through different physical-control balances. Starting a local optimizer from only one design can hide this multiplicity.
Useful safeguards include:
multi-start optimization from physically distinct designs;
continuation from an easier problem or relaxed constraint;
coarse design-space screening before local refinement;
comparison against sequential and plant-only baselines; and
examination of whether different starts converge to the same active constraints and objective.
Convexity is powerful but uncommon¶
For a convex objective over a convex feasible set, every local minimum is global. Some CCD subproblems—such as an unconstrained linear-quadratic controller design for a fixed plant—have favorable structure. The combined plant-control problem usually does not inherit that guarantee.