Bayesian Optimization for Scenario Design

Choosing operating points and profiles is expensive to evaluate; Bayesian optimization finds good scenarios in few costly simulations by balancing exploration and exploitation.

Optimizing an expensive black box

Designing a scenario - the profiles, shape, and actuator schedule that meet targets while respecting stability - means optimizing an objective that costs a full simulation to evaluate. Bayesian optimization (BO) is built for exactly this: it fits a probabilistic surrogate (usually a GP) to past evaluations and uses it to choose the next most-informative point, minimizing the number of expensive runs.

Bayesian optimization loop:
  1. Fit GP surrogate to data { (x_i, y_i) }
  2. Maximize acquisition a(x) to pick next x*
  3. Evaluate expensive objective y* = F(x*)
  4. Augment data; repeat

Expected Improvement acquisition:
  EI(x) = E[ max(0, f_best - f(x)) ]
        = (f_best-mu) Phi(z) + sigma phi(z),  z=(f_best-mu)/sigma

Acquisition functions balance the trade

The acquisition function turns the surrogate's mean and variance into a score for where to sample next. Expected improvement, upper-confidence-bound, and knowledge-gradient all trade exploitation (sample where the mean is good) against exploration (sample where variance is high). This is how BO avoids wasting simulations while still discovering better regions of scenario space.

# one BO iteration (schematic)
gp.fit(X, y)
def acq(x):                      # upper confidence bound
    mu, sd = gp.predict(x)
    return mu + beta*sd          # exploit + explore
x_next = maximize(acq, bounds)   # cheap inner optimization
y_next = expensive_simulation(x_next)
X, y = append(X, x_next), append(y, y_next)

Constraints and honesty

Scenario design is constrained: stability margins, actuator limits, and the operating envelope. Constrained BO models feasibility with its own surrogate and weights the acquisition by the probability of satisfying constraints, so it searches only admissible scenarios. For the burner, the surrogate's variance stays large across the 166-830x regime, so BO is honest that it is exploring an untested space - it proposes candidates for study, not validated operating points, and never optimizes away the four gates.

• Sample-efficient: few expensive simulations to a good design.
• Acquisition balances exploration and exploitation explicitly.
• Constrained BO respects stability and envelope feasibility.
• Burner: wide surrogate variance keeps proposals honest.

BO is the outer optimizer over scenarios; trajectory optimization then refines the time-dependent path within a chosen scenario, and MPC executes it in the loop.