LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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A linear-quadratic transcription with affine constraints is a convex QP

Statement

Consider a finite mesh and an LQR model evaluated by a fixed Runge–Kutta tableau, explicit or implicit, whose stage quadrature weights are nonnegative and whose sampled running and terminal quadratic weights are positive semidefinite. If $A, B, Q, R, x_{d}$ are supplied as almost-everywhere equivalence classes, choose pointwise representatives and state the sampling rule at every stage and quadrature time; the finite QP claim concerns only that resulting sampled system, since an equivalence class alone has no stage value. For an implicit tableau, require that each cell's sampled linear stage-residual matrix is nonsingular and use its unique solution. Suppose every control, path, terminal, initial, and shooting/collocation constraint is specified by finitely many affine equalities or inequalities in the transcription variables. Then the finite transcription of Finite-horizon shooting and Runge–Kutta collocation transcription is a convex quadratic program (QP): its objective is a convex quadratic polynomial and its equality and inequality constraints are affine. This is the finite NLP/QP branch used with the convex models in Conic, LP, and convex QP primal-dual models.

This algebraic claim does not make an $L^{\infty}$ equivalence class a valid input to the pointwise smooth-data procedure. If that procedure is called, its separate pointwise-domain and smoothness hypotheses must also hold; otherwise this lemma concerns only the specified finite sampled equations.

Facts & Assumptions

Given: A finite mesh, chosen pointwise representatives and a sampling rule for any equivalence-class data, linear dynamics, a fixed explicit or implicit Runge–Kutta tableau, nonnegative stage weights, positive-semidefinite sampled quadratic cost weights, nonsingular sampled stage matrices when implicit stages are eliminated, and affine constraints.

[F1]

An explicit RK stage is a finite affine recursion in prior stages. For an implicit linear cell, the slope equations have matrix $M_{i k} = δ_{i k} I - h a_{i k} A (t_{i})$ and right-hand side $A (t_{i}) X_{0} + B (t_{i}) u$ ; if $M$ is nonsingular, their unique solution and the endpoint are affine in $(X_{0}, u)$ (Explicit Runge–Kutta tableaus, stage maps, and failure status, Runge–Kutta tableaux and stage collocation equations).

[F2]

The transcription uses the explicit recursions or solved implicit interval maps for the node/defect equations and uses the displayed quadrature objective at the chosen sample values (Finite-horizon shooting and Runge–Kutta collocation transcription).

[F3]

A convex QP has a positive-semidefinite quadratic objective Hessian and affine equalities/inequalities (Conic, LP, and convex QP primal-dual models, Inequality constrained programs, tangent cones, constraint qualifications, and KKT).

Proof

1.1F1F2giveninductionalgebra

On each interval with linear dynamics, an explicit tableau has affine stages by recursion: the first stage is affine in the start and control, and if previous stages are affine then $X_{i} = X_{0} + h \sum_{k < i} a_{i k} F_{k}$ and $F_{i} = A (t_{i}) X_{i} + B (t_{i}) u$ are affine. For an implicit tableau, the sampled slope equations form the affine system $M K = A X_{0} + B u$ described in [F1]. Nonsingularity makes every stage and endpoint an affine function of $(X_{0}, u)$ ; the algorithm's exact linear stage-solve branch reports failure if this matrix is singular. Thus the endpoint/defect equations and any retained stage equations are affine.

2.1F2F3givenstep 1.1algebra∎

Each quadrature term is a nonnegative scalar weight times a positive-semidefinite quadratic form applied to an affine state/control map, hence has positive-semidefinite Hessian; the terminal form has the same property. A finite sum of such Hessians is positive semidefinite, so the objective is convex quadratic. The remaining constraints are affine by hypothesis; thus [F3] identifies the transcription as a convex QP.

Depends on

Used by

Finite-budget barrier or SQP solve for a transcription NLP Algorithm

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Sources

S. Gros and M. Diehl, Numerical Optimal Control, draft course manuscript, Chapter 13 §§13.1–13.4 (standard reference, not scraped)
W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik 87 (2000) (standard reference, not scraped)