Optimization Under Constraints
A fusion design must be optimized subject to hard limits; the interesting solutions almost always sit on the constraint boundaries.
The constrained problem
Design optimization is rarely about maximizing one number freely. It is about finding the best value of an objective while satisfying constraints: stay below a stress limit, above a breeding requirement, within a space envelope. The best feasible design usually sits where one or more constraints are active, not in the interior.
Types of constraint
- Physics limits, such as stability and density boundaries.
- Engineering limits, such as magnet stress and heat flux.
- Requirement constraints, such as a minimum breeding ratio.
- Geometric limits, such as available space in a compact machine.
Why boundaries matter
Because the optimum sits on constraints, small changes in a limit can move the whole design. This is why sensitivity analysis matters: knowing which constraints bind tells you where reducing uncertainty or relaxing a limit would most improve the design.
Handling constraints numerically
Optimizers handle constraints by penalizing violations, by projecting onto the feasible region, or by working only within it. Which method fits depends on the problem, and the choice affects whether the optimizer finds the true constrained optimum or gets stuck. Constrained optimization runs on surrogates when each evaluation is expensive.
For Kronos
Closing the Hyperion design point is constrained optimization: the closed point satisfies every binding limit at once, and its numbers are consequences of where those limits intersect.
Robust optima
A design sitting exactly on a constraint is fragile if the constraint is uncertain, so uncertainty is included to keep margin against the constraints that are least certain.