Scenario Convex Program (SCP)

• SCP is a convex optimization framework that replaces infinite uncertainty constraints with a finite set of randomized scenarios to approximate robust programs.
• It provides explicit sample complexity bounds and guarantees on violation probabilities through well-defined probabilistic methods and union bounds.
• SCP is widely applied in data-driven control, formal verification, and safe synthesis, enabling reliable decision-making in stochastic systems.

A scenario convex program (SCP) is a convex optimization framework designed to approximate robust or chance-constrained programs involving constraints over uncertainties, by enforcing the constraints only on a finite, randomly sampled subset ("scenarios") of the uncertainty space. It originated within both the robust optimization and stochastic programming literatures as a computationally tractable surrogate for convex programs whose feasible set is defined by an infinite or probabilistically-constrained family of inequalities. SCPs have become foundational tools for data-driven control, safe synthesis, formal verification, abstraction-based design, and compositional analysis in both continuous and discrete stochastic systems.

1. Mathematical Formulation and Core Principles

Consider a robust convex program (RCP) of the form:

$$\min_{x\in X} \ c^\top x \quad \text{subject to} \quad f(x,d) \le 0 \quad \forall d\in D,$$

where $D$ is the uncertainty set (which may be infinite or high-dimensional) and $f(x,d)$ is convex in $x$.

An SCP replaces the universal quantifier with $N$ independent samples $\{d_i\}_{i=1}^N$ drawn from a probability measure on $D$, yielding:

$$\min_{x\in X} \ c^\top x \quad \text{subject to} \quad f(x,d_i) \le 0,\quad i = 1, ..., N.$$

This convex program is tractable as long as $f$ and $X$ are convex, and inherits its feasibility and optimality properties from the sampled representation of the uncertainty.

In the context of chance-constrained programs (CCP), i.e.,

$$\min_{x\in X} c^\top x \quad \text{subject to} \quad \mathbb{P}\big[ f(x,d) \le 0 \big] \ge 1-\epsilon,$$

SCP provides a rigorous probabilistic relaxation for satisfying constraints with a guaranteed violation probability, even when the distribution of $d$ or the set $D$ is only partially known (Esfahani et al., 2013, Schildbach et al., 2012).

2. Sample Complexity and Out-of-Sample Guarantees

The primary theoretical results establish explicit, dimension-dependent relationships between the number of scenarios $N$, the violation probability $\epsilon$, the confidence $1-\beta$, and the structure of constraints.

The canonical bound is:

$$\sum_{i=0}^{d-1} \binom{N}{i} \epsilon^i (1-\epsilon)^{N-i} \le \beta,$$

where $d$ is the number of decision variables (or support rank in specialized setups) (Schildbach et al., 2012, Esfahani et al., 2013, Altiner et al., 2024).

For convex programs, with high probability, the solution $x_N^\star$ of the SCP is feasible for the chance-constrained program with violation probability at most $\epsilon$:

$$\mathbb{P}^{N} \left\{ \mathbb{P}(f(x_N^\star,d)>0) > \epsilon \right\} \le \sum_{i=0}^{d-1} \binom{N}{i} \epsilon^i (1-\epsilon)^{N-i}.$$

This result extends via union bounds to multiple chance constraints, and can be improved by exploiting the notion of support rank (number of "active" constraints) (Schildbach et al., 2012). Sample sizes can be dramatically reduced in multi-stage or decomposed problems where constraint functions are lower-dimensional.

In the presence of empirical expectation approximations (e.g., for stochastic systems), additional concentration inequalities (Chebyshev, Chernoff) yield sample complexity bounds balancing the number of noise samples $M$, slack margins $\mu$, and overall confidence (Lavaei et al., 2022, Salamati et al., 2021).

3. Variants and Extensions: Discarding, Validation, Non-convexity

SCP theory has advanced to embrace several extensions:

• Sampling-and-Discarding: Additional "bad" scenarios may be removed a posteriori (e.g., for cost improvement), with exact binomial-tail violation bounds reflecting the total number of discarded constraints (Romao et al., 2020, Schildbach et al., 2012). The cascade removal scheme is shown to be tight via compression learning bounds.
• A Posteriori Validation: By leveraging both the count of support constraints and measured empirical violations on new validation data, one can refine the violation probability and achieve substantially lower sample complexity for fixed confidence (Chen et al., 2023).
• Non-convex Scenario Programs: SCP can be extended to certain unions of convex feasible sets, notably in mixed-integer programs and structured non-convex programs, with linear-in-union sample complexity via union-bound arguments (Esfahani et al., 2013, Salamati et al., 2021).

4. Integration in Data-Driven Control, Synthesis, and Verification

SCP is systematically applied as a tractable surrogate for RCPs in synthesis and verification across diverse domains. In abstraction-based control synthesis, SCP is used to fit growth bounds over sampled trajectories of an unknown system, which then inform the construction of finite abstractions and symbolic controllers that fulfill temporal logic requirements with probabilistic guarantees (Kazemi et al., 2022).

In safe controller synthesis and barrier-certificate construction for unknown stochastic and deterministic systems, SCP gives explicit control barrier function parameters, with confidence-controlled guarantees on safety properties, even in the presence of unknown transition kernels or unmodeled noise (Salamati et al., 2021, Chen et al., 2023). When combined with data-driven compositionality conditions, SCP delivers stability and safety certificates for interconnected nonlinear networks, with scalability improvements (e.g., linear sample complexity in the number of subsystems) (Lavaei et al., 23 Dec 2025).

5. Computational Strategies and Algorithmic Details

SCPs are convex programs typically solved via standard interior-point or first-order solvers (CVX, MOSEK, etc.). When the scenario program is an SDP (e.g., pole placement via LMI relaxations) (Altiner et al., 2024), tailored SDP solvers (YALMIP + SeDuMi) are used. For compositional SCP formulations, linear programming suffices (Lavaei et al., 23 Dec 2025).

Key steps are:

• Sample $N$ scenarios from the uncertainty space.
• Formulate and solve the SCP for decision variable(s) under convex (possibly LMI) constraints for each scenario.
• (Optionally) perform discarding of constraints or a posteriori validation on fresh data.
• Extract controller parameters, abstractions, certificates, or Lyapunov/storage function coefficients.

Algorithms may integrate sparse automatic differentiation (for dynamical linearization) (Yarndley, 23 Nov 2025) or extreme-value fitting (for Lipschitz constant estimation) (Kazemi et al., 2022).

6. Applications Across Domains and Practical Impact

SCP has enabled formal abstraction and safe control synthesis for unknown stochastic systems (Lavaei et al., 2022, Kazemi et al., 2022), robust pole placement under contact-model uncertainty in dexterous manipulation (Altiner et al., 2024), sequential convex programming for nonconvex spacecraft trajectory optimization (Yarndley, 23 Nov 2025), compositional and scalable stability analysis of nonlinear networks (Lavaei et al., 23 Dec 2025), and data-driven safe control for both stochastic and deterministic settings (Salamati et al., 2021, Chen et al., 2023).

Performance bounds of SCP-based controllers and certificates are validated in case studies ranging from multi-fingered grasping under contact-location uncertainty, to compositional GAS certification in interconnected homogeneous networks, jet engine compressor abstraction, and resource-sharing linear programs. In each domain, the scenario approach allows practitioners to rigorously quantify uncertainty-induced performance degradation or safety violation probability in terms of sample size and support rank, without requiring explicit models or full distributional specification.

7. Limitations, Scalability, and Ongoing Research

Sample complexity typically scales polynomially with the effective dimensionality (support rank, number of decision variables) and inversely with the desired violation probability. In high-dimensional settings, the curse of dimensionality may render classic SCP bounds pessimistic; compositional approaches and posterior validation methods have significantly reduced sample requirements in practice (Schildbach et al., 2012, Lavaei et al., 23 Dec 2025, Chen et al., 2023).

SCP theory underpins the empirical observation that the number of support constraints is often far below the program dimension, enabling tight confidence bounds. Some variants (e.g., cascade discarding) deliver exact bounds that generalize classical scenario results (Romao et al., 2020). Extensions to unions of convex sets (non-convex SCPs), mixed-integer variants, strong performance guarantees for stochastic abstraction, and compositional analysis of large-scale networks remain active research directions, as evidenced in recent work (Esfahani et al., 2013, Lavaei et al., 23 Dec 2025).

The scenario convex program now defines one of the primary pathways from rich trajectory data to reliable, scalable, and formally-correct decision-making under uncertainty.