Performative scenario optimization solutions converge to classical stochastic programming solutions as the strength of the decision-feedback effect decreases, with convergence rate bounded by the Lipschitz modulus of the distribution map.

Let P_θ(x) denote a decision-dependent (performative) distribution map where θ ∈ [0,1] parameterizes the strength of decision-feedback (θ=0 recovers the classical, decision-independent distribution P_0), and let ε(θ) = sup_x W(P_θ(x), P_0) be the Lipschitz modulus of the distribution map with respect to the decision variable x, measured in Wasserstein distance W(·,·). Define x*(θ) as the performatively stable (or optimal) solution of the scenario optimization problem under P_θ, and x_0 the classical stochastic programming solution under P_0. The falsifiable claim is: (1) ‖x(θ) − x_0‖ → 0 as θ → 0, and (2) there exist constants C > 0 and a monotone function L(θ) (the empirical Lipschitz modulus of the map θ ↦ P_θ) such that ‖x(θ) − x*_0‖ ≤ C·L(θ) for all θ in a neighborhood of 0, with L(θ) → 0 as θ → 0 at a rate no slower than O(θ) under strong convexity of the objective and joint strong monotonicity assumptions standard in performative prediction (Perdomo et al. 2020 framework).

Minimum viable test: synthetic performative scenario optimization problems (portfolio selection with strategic price response, newsvendor with demand-shift feedback) where P_θ is constructed analytically (e.g., mean/covariance shift linear in θ·x), permitting closed-form or near-closed-form x_0 and controllable ground-truth L(θ). Sweep θ ∈ {0, 0.01, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0}, solve performative scenario optimization via repeated retraining (RRM) or performative gradient descent, compare to classical SAA (sample average approximation) solution at θ=0, measure ‖x(θ)−x*_0‖_2 and regress against measured L(θ).

# Pseudocode: Performative Scenario Optimization Convergence Test

def classical_SAA(P0, N=10000):
    scenarios = sample(P0, N)
    x0_star = solve_convex_program(scenarios)
    return x0_star

def performative_RRM(theta, P0, g, max_iter=50, tol=1e-6):
    x_t = initialize_x()
    for t in range(max_iter):
        P_theta_xt = P0 + theta * g(x_t)   # decision-dependent shift
        scenarios = sample(P_theta_xt, N=10000)
        x_next = solve_convex_program(scenarios)
        if norm(x_next - x_t) < tol:
            break
        x_t = x_next
    L_theta = wasserstein_distance(P_theta_xt, P0)  # Sinkhorn approx
    return x_t, L_theta

def run_experiment():
    x0_star = classical_SAA(P0)
    results = []
    for theta in [0, 0.01, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0]:
        for seed in range(30):
            x_theta, L_theta = performative_RRM(theta, P0, g, seed=seed)
            gap = norm(x_theta - x0_star)
            results.append((theta, L_theta, gap, seed))
    # regression: log(gap) ~ beta * log(L_theta) + log(C)
    beta, logC, R2 = fit_loglog(results)
    return results, beta, logC, R2