Source-linked AI summary
Generalized Semi-Infinite Programming for Robust Optimal Control with Decision-Dependent Uncertainty
J. Wehbeh, E. C. Kerrigan, E. Scaccia
TL;DR
The paper addresses GSIPs and robust optimal-control problems whose uncertainty depends on decisions, avoiding restrictive structure and specialized global optimization. It reformulates them as existence-constrained SIPs solved by adaptive discretization, with convergence under stated assumptions and feasibility demonstrated for state-dependent satellite uncertainty.
Problem
GSIPs with decision-dependent uncertainty lack broadly applicable methods, while existing approaches impose restrictive structure or require poorly scaling global optimization for control problems.
Method
The framework transforms suitably regular GSIPs into existence-constrained SIPs with decision-independent uncertainty and solves them using adaptive discretization with standard nonlinear-programming solvers.
Results
The method converges under the theorem’s assumptions, and the satellite’s state-dependent formulation is feasible on sampled and worst-edge trajectories while the static surrogate is infeasible.
Takeaways & Limitations
Decision-dependent uncertainty can be modeled directly within a standard SIP cutting-plane framework, extending robust optimal control to state-dependent uncertainty.
Takeaways & Limitations
The convergence guarantee requires regularity, feasibility, and globally solved finite subproblems, which are not guaranteed for the paper’s nonconvex programs.
Abstract
from arXiv · showhide
Generalized semi-infinite programs (GSIPs) arise in robust optimal control whenever the admissible uncertainty depends on the state or controls. Existing GSIP methods either impose restrictive structural assumptions or require global optimization that scales poorly to control problems. We present a general framework that reformulates any GSIP with mild regularity as an existence-constrained semi-infinite program, smoothing its disjunctive feasibility condition into differentiable existence constraints over a fixed index superset. The resulting program is solved by established adaptive discretization (cutting-plane) methods using only off-the-shelf nonlinear-programming solvers, and converges under standard assumptions. Treating the state trajectory as part of the uncertainty extends the framework to robust nonlinear optimal control with state-dependent uncertainty. We demonstrate it on a nonconvex benchmark GSIP and a satellite de-tumbling problem with dynamically varying inertia.
A. Background and Motivation S
Semi-infinite programs impose infinitely many constraints over an index set, while GSIPs allow that set to depend on operational decisions. This decision dependence complicates the direct use of classical SIP methods.
- Semi-infinite programs have finite-dimensional decisions but infinitely many constraints indexed over a fixed set.
- Generalized SIPs arise when the uncertainty index set depends directly on controls or indirectly on system states.
- GSIPs are not always tractable with classical SIP methods, motivating conversion to existence-constrained SIPs.
B. State of the Art
Existing approaches to state-dependent uncertainty are limited in coverage or computational form. The paper instead targets general GSIP techniques that can reuse standard SIP machinery without requiring global optimization in every case.
- Explicit state-dependent uncertainty methods are uncommon and mostly learning-based, with adaptive sliding-mode control as a non-learning exception.
- Only one cited data-driven method guarantees robust constraint satisfaction, while the cited alternatives do not optimize directly over the worst case.
- Direct state-dependent SIP methods include restrictive polynomial-polyhedral approaches, while global discretization or branch-and-bound methods address nonconvexity at higher computational cost.
- Existing transformations either impose restrictive global assumptions or provide more limited local transformations, unlike the proposed existence-constrained approach.
C. Contributions
The paper develops a general GSIP-to-SIP framework, integrates it with adaptive discretization, extends it to nonlinear robust optimal control, and validates it on benchmark and satellite problems.
- A refined transformation converts any suitably regular GSIP into an existence-constrained SIP by smoothing disjunctive feasibility constraints.
- The transformed problem integrates with adaptive discretization and converges under standard assumptions when each subproblem is solved globally.
- A broad class of nonlinear robust optimal control problems with decision-dependent uncertainty is cast as GSIPs and solved with the same machinery.
- The framework is validated on a nonconvex GSIP benchmark and a satellite de-tumbling problem with inertial uncertainty.
II. GENERALIZED SEMI-INFINITE PROGRAMMING
Under compactness, a uniquely determined index component, and smoothness, the GSIP is reformulated over a fixed index superset. A smoothed existence constraint preserves feasibility exactly under a density qualification.
- Problem structure: The formulation separates the index into a uniquely determined component p and a free component q ranging over a compact fixed superset.
- Assumptions: Assumption 1 requires compactness, a unique continuous root for p, containment of feasible indices, and neighborhood-defined functions.
- Assumptions: Smoothness of f, g, h, and r makes the objective and finite subproblems differentiable.
- Reformulation: The GSIP is equivalent to a semi-infinite program over the fixed set ¯Q after reducing each index through the root equation.
- Exactness: The non-strict relaxation is exact when strictly admissible indices are dense, but can fail if decision dependence collapses admissibility onto its boundary.
- Reformulation: The disjunctive feasibility condition is smoothed into differentiable existence constraints using a continuous simplex representation.
III. ADAPTIVE DISCRETIZATION FOR GENERALIZED SIPS
The method reformulates the existence-constrained problem as a standard semi-infinite program and solves it by adaptive discretization with separation-oracle updates. Under stated assumptions and globally solved finite subproblems, the algorithm converges to an optimal solution.
- Standard SIP reformulation: The existence-constrained formulation is equivalent to a standard semi-infinite program over the fixed compact index set ¯Q.The reformulation uses Gi(z, q), defined as the minimum of the robust constraint functions at the uniquely determined implicit variable.
- Master problem: For a finite working set Q, the master problem is exact at retained indices but relaxes the full problem and lower-bounds its optimal value.Simplex multipliers make the existence constraints explicit while preserving differentiability of the finite nonlinear program.
- Separation oracle: The separation oracle computes each worst-case violation by maximizing Gi over ¯Q through a finite differentiable nonlinear program whose maximum is attained.The oracle does not need simplex multipliers because its epigraph formulation represents the maximum of Gi with smooth constraints.
- Adaptive discretization: Each iteration adds the worst-case discretization index to Q and stops when the largest violation is nonpositive.This converts the generalized problem into a cutting-plane procedure on the equivalent standard SIP.
IV. STATE-DEPENDENT UNCERTAINTIES IN ROBUST OCPS
The framework represents robust optimal control with state- and decision-dependent uncertainty as a generalized semi-infinite program. It treats the state trajectory as part of the uncertainty while using dynamics and auxiliary equalities to determine state components from a reduced free index.
- Problem formulation: A discrete-time robust control problem minimizes worst-case cost over disturbances and state trajectories constrained by dynamics and state-dependent admissibility conditions.The uncertainty includes x and w, while terminal or boundary requirements are represented by robust control constraints.
- Regularity assumptions: Compactness, unique solvability, interior containment, and smoothness assumptions make the control problem fit the generalized SIP framework.The assumptions ensure the functions are defined around evaluated candidates and support the required differentiability and worst-case searches.
- Uncertainty structure: The uncertainty is partitioned as v=(x, pc, qc), with (x, pc) uniquely and continuously determined by the dynamics and equality constraints for each free index qc.This supplies the determined part required by the GSIP reformulation.
- Interpretation: Although the state is treated as uncertain, uniqueness means the inner maximization remains the usual worst case over disturbances evaluated at the state solving the dynamics.The state trajectory is therefore included in the uncertainty representation without changing the disturbance-based interpretation of the inner maximization.
- Algorithmic integration: Under Assumption 2 and Theorem 2's conditions, the control problem can be written as a differentiable GSIP and solved to optimality by Algorithm 1.The same adaptive discretization procedure operates over a finite working set of qc values with state and auxiliary-variable copies at retained points.
V. NUMERICAL RESULTS
The numerical implementation uses local nonlinear optimization with multistart rather than the global subproblem solves required by the convergence theorem.
- Numerical methodology: Because the numerical problems are nonconvex, the implementation uses fast local optimizers with multistart instead of guaranteed globally optimal subproblem solutions.The implementation uses JuMP for transcription and Ipopt as the local optimizer, with randomized initial guesses for discretization points.
A. GSIP Example
On a nonlinear, nonconvex benchmark GSIP, Algorithm 1 converges rapidly across repeated runs and is substantially faster than a competing globally oriented method.
- GSIP benchmark: 5–6 iterations and 38.96 ms on average were required across 10,000 runs for Algorithm 1 to reach tolerance 10−4.The benchmark is nonlinear and nonconvex in its constraints, with ¯Q=[0,π] and q=v.
- GSIP benchmark: 23.46 s was required by the method of on the same problem, with additional global optimality guarantees.The comparison used older hardware, so the timing difference is not a controlled hardware comparison.
B. Satellite De-tumbling
The satellite de-tumbling problem models vibration-induced inertia as uncertainty dependent on angular rates and solves the resulting GSIP through discretized dynamics and inertia relations. The state-dependent formulation is feasible, while a state-independent surrogate is infeasible.
- Problem formulation: The satellite is modeled as approximately planar, with vibration-induced inertial uncertainty depending only on angular rates and fixed Izz determined by the perpendicular-axis theorem.Off-diagonal inertia terms are neglected.
- GSIP solution: The rate-independent floor ε > 0 keeps admissible inertia intervals nonzero, making the non-strict relaxation exact; ε = 0 can admit a non-robust control vacuously.At ε = 0, driving ωk toward zero collapses the interval to {Ī}.
- Problem formulation: The discrete-time model propagates unit quaternion attitude and angular velocity using the current inertia vector and applied control torque.The dynamics are represented compactly by F(qk, ωk, τk, Ik).
- Optimization problem: The control problem minimizes the sum of squared torques while requiring the satellite to approach a reference quaternion within ∆θmax by step N.The admissible torque trajectories form a compact box whose bounds are inactive at the solution.
- GSIP solution: The GSIP reformulation treats (Ixx, Iyy) as the per-step free uncertainty and (x, Izz) as the determined part, with dynamics and Izz = Ixx + Iyy enforced per point.The compact inertia superset is finite because the horizon and state space are compact.
- Results: The state-independent surrogate is infeasible under the imposed angular-rate bounds, whereas the state-dependent formulation satisfies the robustness requirement.The surrogate bounds |ωx| and |ωy| by 0.5 rad/s.
VI. CONCLUSIONS
The paper transforms generalized SIPs into existence-constrained SIPs and applies standard adaptive discretization to robust nonlinear control with decision-dependent uncertainty. The satellite study shows that modeling uncertainty dependence directly can determine feasibility, while the convergence guarantee remains conditional.
- Contributions: The proposed transformation recasts a suitably regular GSIP as an existence-constrained SIP with decision-independent uncertainty.It smooths disjunctive feasibility constraints into the transformed formulation.
- Contributions: The reformulation uses standard adaptive discretization machinery rather than a solver tailored to generalized SIPs.The same machinery applies to the transformed problem.
- Satellite study: Modeling decision dependence directly makes the satellite formulation feasible, whereas the static state-independent surrogate is infeasible.The state-dependent formulation meets the bound on sampled and worst-edge trajectories.
- Limitations: Future work includes relaxing uniqueness of the determined uncertainty part, bounding local-solve suboptimality and discretization size, and extending the framework to probabilistic uncertainty and larger problems.These are stated as natural next steps.