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Performance Bounds for the Scenario Approach and an Extension to a Class of Non-convex Programs
Peyman Mohajerin Esfahani, Tobias Sutter, John Lygeros
TL;DR
The paper asks how tractable scenario programs can provide performance guarantees for robust and chance-constrained optimization problems that may be computationally intractable. It develops probabilistic objective-value bounds over general metric uncertainty spaces, extends them to selected non-convex programs, and resolves a related measurability issue. The resulting framework covers binary decision variables and is illustrated on benchmark and fault-detection applications.
Problem
Robust and chance-constrained programs can be intractable, and existing scenario-program theory provides limited connection to their objective performance.
Method
The paper bounds the tail probability of worst-case constraint violation and uses perturbation results to connect scenario optimal values with robust and chance-constrained optimal values.
Results
The paper provides confidence bounds for objective performance in robust and chance-constrained programs and extends the results to a class of non-convex programs allowing binary decision variables.
Takeaways & Limitations
The framework supplies scenario-based objective-performance guarantees beyond feasibility for uncertainty represented in general metric spaces.
Takeaways & Limitations
The confidence analysis may require a strictly feasible solution, and the sample count can grow exponentially as the desired precision increases with uncertainty dimension.
Abstract
from arXiv · showhide
We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the optimal value of SCP to the optimal values of RCP and CCP in which the uncertainty takes values in a general, possibly infinite dimensional, metric space. We then extend our results to a certain class of non-convex problems that includes, for example, binary decision variables. In the process, we also settle a measurability issue for a general class of scenario programs, which to date has been addressed by an assumption. Finally, we demonstrate the applicability of our results on a benchmark problem and a problem in fault detection and isolation.
1. Introduction
The paper studies scenario convex programs as tractable approximations for robust and chance-constrained optimization under uncertainty. It focuses on connecting scenario solutions to objective performance, extending results to infinite-dimensional uncertainty, and resolving a measurability issue.
- Motivation: Robust convex programs require constraints to hold for all uncertainty realizations, while chance-constrained programs impose probabilistic constraint satisfaction.Some robust convex programs are tractable, but others are computationally intractable.
- Scenario approximation: Scenario convex programs replace potentially infinite uncertainty constraints with finitely many sampled constraints to obtain tractable approximations.The central question is how many samples suffice for good feasibility or objective performance.
- Research gap: Existing scenario-program research mainly addresses feasibility, whereas objective performance and its connection to the original programs remain less settled.Earlier performance work involved optimal constraint removal, which is generally computationally intractable.
- Contributions: The paper extends worst-case-violation performance analysis from finite-dimensional Euclidean uncertainty to possibly infinite-dimensional metric spaces.The analysis is motivated by applications including control with saturation constraints, fault detection and isolation, and approximate dynamic programming.
- Contributions: The authors establish probabilistic links between scenario-program optimal values and the optimal values of robust and chance-constrained programs.The stated assumptions concern measurability in uncertainty and lower semicontinuity in decision variables.
2. Problem Statement
The problem statement formalizes scenario programs over a metric uncertainty space and distinguishes feasibility from objective performance. It motivates studying probabilistic links between sampled optimization and robust or chance-constrained formulations while addressing measurability of the scenario optimizer.
- Formal setting: The paper considers a measurable function on a metric uncertainty space that is convex in the decision variables and bounded in uncertainty for each decision.The decision set is compact and convex, and the uncertainty uses its Borel σ-algebra.
- Scenario program: The scenario program uses N independent and identically distributed uncertainty samples and imposes one constraint for each sample.Its optimizer and optimal value are random because they depend on the sampled uncertainties.
- Measurability: The analysis initially assumes that the scenario optimizer induces a Borel measurable mapping, while later results address this issue for a broader class of programs.The stated measurability assumption concerns the mapping generated by the scenario optimizer.
- Motivation: Scenario convex programs are tractable even when the corresponding robust or chance-constrained problems are NP-hard.This tractability motivates seeking theoretical links from scenario solutions to the original formulations.
- Feasibility and performance: Feasibility theory provides sample-size guarantees for chance-constrained programs, but the study focuses on probabilistic bounds for optimal values.The recalled result guarantees chance-constraint feasibility with probability at least 1 −β when N is sufficiently large.
- Research focus: The paper notes that no clear connection is known between robust-program feasibility and the solution of a scenario program.It therefore emphasizes the objective-performance perspective rather than only feasibility.
3. Probabilistic Objective Performance
This section develops probabilistic objective-performance bounds connecting scenario-program solutions to robust and chance-constrained programs. It introduces uniform level-set bounds, derives a priori and a posteriori confidence intervals, extends the analysis to infinite-dimensional uncertainty spaces, and addresses optimizer measurability.
- Uniform level-set bounds: Uniform level-set bounds control the tail probability of worst-case constraint violation and support objective-performance guarantees.The paper defines a ULB uniformly over decisions and uses it to relate violation levels to relaxed robust programs.
- Confidence intervals: The RCP confidence result bounds the scenario program’s optimal value using a relaxed robust program under a Slater-point assumption.The bound depends on the ULB, the sample size, and the Lipschitz constant associated with robust-value relaxation.
- Confidence intervals: The CCP result provides both a priori and a posteriori confidence intervals for the scenario program’s optimal value.The a posteriori interval uses the dual optimizer, while the parameter ε is fixed by the CCP data rather than tuned as in the RCP result.
- Uniform level-set bounds: Proposition 3.8 constructs admissible bounds for possibly infinite-dimensional uncertainty spaces, although its assumptions implicitly require boundedness.The proposition generalizes an earlier finite-dimensional result; under the stated metric-growth conditions, the required sample size can grow exponentially as ε^-nd.
- Feasibility and measurability: The SCP optimizer can almost surely be infeasible for the robust program when worst-case scenarios have measure zero.Scenario samples may neglect the scenarios attaining the worst-case constraint value, so feasibility and objective performance require separate assessment.
- Feasibility and measurability: A lexicographic two-stage scenario program ensures that the optimizer is a measurable singleton for a broad class of programs.The construction also addresses the lack of an a priori measurability guarantee for the ordinary SCP optimizer.
4. Extension to a Class of Non-Convex Programs
The paper extends scenario-program feasibility and objective-performance guarantees to a class of non-convex programs, including mixed-integer problems with binary variables. It models these problems through multiple convex subprograms sharing uncertainty samples and derives confidence bounds whose sample growth depends logarithmically on the number of subprograms.
- Problem class: The non-convex formulation seeks an optimal solution feasible for at least one of multiple subprograms sharing the same uncertainty space and probability measure.Each subprogram uses its own program data, while the uncertainty model is common across subprograms.
- Problem class: Chance-constrained mixed-integer programs with binary variables fit this framework by enumerating binary selections as separate convex subprograms.For ℓ binary variables, the construction sets m := 2^ℓ and uses identical feasible sets and violation levels across selections.
- Feasibility guarantee: Theorem 4.1 guarantees that the optimizer of the scenario program is feasible for the non-convex program with probability at least 1 − β when N ≥ eN(ε⃗, β).The result extends the scenario feasibility guarantee from convex programs to the non-convex subprogram formulation.
- Sample growth: The number of subprograms contributes linearly to the aggregate confidence level when the subprogram violation levels are equal.The paper describes the resulting confidence level as β^m when β is the confidence level assigned to each subprogram.
- Sample growth: For mixed-integer programs, the required sample count grows linearly with the number of binary variables, while the dependence on the number of subprograms enters logarithmically.The paper contrasts its ε^-1 log(m) growth with a prior bound scaling as ε^-1 log(αkε^-1).
- Objective guarantees: Theorem 4.3 extends robust and chance-constrained objective confidence intervals to the non-convex program class through corresponding subprogram intervals.The proof applies the convex-program interval results separately to each subprogram.
5. Simulation Results
The paper illustrates its theoretical performance bounds through an analytical benchmark and a fault-detection application. The examples compare confidence intervals with empirical behavior and apply the method to a non-convex FDI design problem.
- 5.1. Example 1: Quadratic Constraint via Infinite Hyperplanes.: The first example replaces infinitely many hyperplane constraints with a quadratic constraint and provides an analytically solvable benchmark.The decision variables lie in [0,1]^2, while uncertainty is uniformly distributed over [0,2π].
- 5.1. Example 1: Quadratic Constraint via Infinite Hyperplanes.: The benchmark computes analytical optimal values for the robust and chance-constrained formulations and compares them with scenario-program results.
- 5.1. Example 1: Quadratic Constraint via Infinite Hyperplanes.: The empirical confidence interval is defined as the smallest interval containing the robust optimal value in at least m experiments.
- 5.1. Example 1: Quadratic Constraint via Infinite Hyperplanes.: The theoretical confidence interval contains the empirical interval, while the a posteriori interval is tighter than the a priori interval in both Figure 4 cases.
- 5.2. Example 2: Fault Detection and Isolation.: The power-network study depicts disturbance realizations, intrusion onset at t = 10, residual energy, and the threshold γ⋆ + 0.0005 over 15 seconds.The residual-energy plot uses the final T = 4 seconds and compares the response with the β = 0.01 threshold.
6. Conclusion and Future Direction
The paper presents probabilistic objective-performance bounds for robust and chance-constrained programs solved through scenario programs, then extends them to selected non-convex programs. It identifies unresolved directions involving uncertainty-dependent upper bounds and the relationship between Slater constants and dual optimizers.
- Conclusion: The proposed bounds use the tail probability of the scenario solution’s worst-case constraint violation together with convex-optimization perturbation results.
- Conclusion: The work claims the first confidence bounds for objective performance of robust and chance-constrained programs based on scenario programs.
- Conclusion: The results extend to a class of non-convex programs that allows binary decision variables.
- Future direction: Future work targets deriving meaningful upper-bound functions whose form depends on the uncertainty set and constraint functions.
- Future direction: The authors also propose studying whether scenario-program dual optimizers generally approximate performance better than the constant LSP.
A. Appendix: Technical Proofs
The appendix supplies proof ingredients for the performance results, including worst-case-violation bounds, perturbation arguments, and measurability of optimization mappings. It also establishes measurability of scenario-program optimizers through measurable set-valued mappings and uniqueness.
- Worst-case violation bounds: A sequence approaching the supremum over uncertainty supports the upper-bound-function argument used in the technical proofs.
- Perturbation analysis: Under strong duality, the robust perturbation function is Lipschitz with constant bounded by the Slater-point constant LSP.
- Measurability: The optimization value mapping J is measurable on lower-semicontinuous functions equipped with the infinite norm and Borel σ-algebra.
- Measurability: The set-valued optimizer mapping is measurable, and strict convexity makes the optimizer a singleton.
- Scenario-program measurability: The scenario optimizer is measurable because it is the composition of the measurable maximum mapping with the measurable optimizer mapping.
B. Appendix: Details of Example 2
The appendix describes the two-area power network used in the fault-detection example as a system of nonlinear ordinary differential equations.
- System model: The two-area power network is modeled by a set of nonlinear ordinary differential equations.
B.1. Mathematical model description.
The model describes a two-area system with a seven-dimensional state vector and fixed physical parameters, where the disturbance enters through the first area's load.
- The system state is represented by a vector in R^7.
- Both areas use the same specified physical parameters, including Tchi = 5 sec, Hi = 6.5 sec, and f0 = 50 Hz.
- The disturbance signal d affects the first area's load, while the second area's load disturbance is identically zero.
B.2. Lipschitz constant of the mapping d 7→Qd.
The mapping from disturbance signals to outputs is analyzed through a two-step decomposition, with a Lyapunov-like approach used to obtain a less conservative Lipschitz estimate than classical ODE continuity arguments.
- The disturbance-to-output mapping is decomposed into d 7→E(X) followed by E(X) 7→Qd.
- A Lyapunov-like approach is used to approximate the Lipschitz constant of d 7→E(X) more efficiently than standard continuity bounds based on Gronwall’s inequality.
- The Lyapunov condition bounds the derivatives of V along paired state trajectories by −κV(X, eX) + ρ.
- The search is restricted to quadratic functions V(X, eX) = (X − eX)⊺Q(X − eX), with Q constrained by a positive-semidefinite matrix inequality.
- For the example, LMIs yield a local Lyapunov function valid over specified bounded ranges of frequency, angle, and power deviations.