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Credible autocoding of convex optimization algorithms

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Abstract

The efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety-critical roles. There is a considerable body of mathematical proofs on on-line optimization algorithms which can be leveraged to assist in the development and verification of their implementation. In this paper, we demonstrate how theoretical proofs of real-time optimization algorithms can be used to describe functional properties at the level of the code, thereby making it accessible for the formal methods community. The running example used in this paper is a generic semi-definite programming solver. Semi-definite programs can encode a wide variety of optimization problems and can be solved in polynomial time at a given accuracy. We describe a top-down approach that transforms a high-level analysis of the algorithm into useful code annotations. We formulate some general remarks on how such a task can be incorporated into a convex programming autocoder. We then take a first step towards the automatic verification of the optimization program by identifying key issues to be addressed in future work.

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Notes

  1. The terms “interior-point method” and “interior-point algorithm” are used interchangeably in this paper.

  2. In this case, the condition that p is transformable to a symmetric matrix P is guaranteed by the way SDP problems in control are formulated (Gahinet and Apkarian 1994; Boyd 1994).

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Acknowledgments

The authors would like to thank the French National Research Agency (ANR) for its support under ASTRID Project VORACE (Vérification de l’Optimisation Rapide Appliquée à la Commande Embarquée), the National Science Foundation under CNS Grants 1135955 “CPS: Medium: Collaborative Research: Credible Autocoding and Verification of Embedded Software (CrAVES)”, and 1446758 “CPS: Synergy: Collaborative Research: Semantics of Optimization for Real Time Intelligent Embedded Systems (SORTIES)”, as well as the Army Research Office under MURI Award W911NF-11-1-0046.

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Appendix

Appendix

The function \({\texttt{vecs}}\) is similar to the standard vectorization function but specialized for symmetric matrices. It is defined as, for \(1\le i < j \le n\) and \(M \in {\mathbb {S}}^{n},\)

$$\begin{aligned} \displaystyle {{\mathrm{vecs}}}\, {M} = \begin{bmatrix} M_{11}, \ldots , \sqrt{2} M_{ij}, \ldots , M_{nn} \end{bmatrix} ^{\text {T}}. \end{aligned}$$
(47)

The factor \(\sqrt{2}\) ensures the function \({\texttt{vecs}}\) preserves the distance defined by the respective inner products of \({\mathbb {S}}^{n}\) and \(\mathbb {R}^{\frac{n \left( n+1\right) }{2}}.\) The function \({\texttt{mats}}\) is the inverse of \({\texttt{vecs}}.\) The function \({\texttt{krons}},\) denoted by the symbol \(\otimes _{sym},\) is similar to the standard Kronecker product but specialized for symmetric matrix equations. It has the property

$$\begin{aligned} \displaystyle \left( Q_{1} \otimes _{sym}Q_{2} \right) {{\mathrm{vecs}}}{(M)} = {{\mathrm{vecs}}}{\left( \frac{1}{2} \left( Q_{1} M Q_{2}^{\text {T}} + Q_{2} M Q_{1}^{\text {T}} \right) \right) }. \end{aligned}$$
(48)

Let \(Q_{1}=TZ\) and \(Q_{2}=T_{inv}\) and \(M=\varDelta X,\) we get

$$\begin{aligned} \displaystyle \left( TZ \otimes _{sym}T_{inv} \right) {{\mathrm{vecs}}}{(\varDelta X)} = {{\mathrm{vecs}}}{\left( \frac{1}{2} \left( TZ \varDelta X T_{inv} + T_{inv} \varDelta X ZT\right) \right) }. \end{aligned}$$
(49)

Additionally, let \(Q_{1}=T,\,Q_{2}=XT_{inv},\) and \(M=\varDelta Z,\) we get

$$\begin{aligned} \displaystyle \left( T \otimes _{sym}XT_{inv} \right) {{\mathrm{vecs}}}{(\varDelta Z)} = {{\mathrm{vecs}}}{\left( \frac{1}{2} \left( T \varDelta Z XT_{inv} + T_{inv} X \varDelta Z T\right) \right) }. \end{aligned}$$
(50)

Combining (49) and (50), we get exactly the left hand side of the third equation in (24). Given a \(\varDelta Z,\) we can compute \(\varDelta X\) by solving \(A\delta x=b\) for \(\delta x\) where

$$\begin{aligned}&A=\left( TZ \otimes _{sym}T_{inv} \right) , \\&\varDelta X = {{\mathrm{mats}}}{\left( \delta x,\,n\right) }, \\&b={{\mathrm{vecs}}}{\left( \sigma \mu I - T_{inv} X T_{inv} \right) } - \left( T \otimes _{sym}XT_{inv} \right) {{\mathrm{vecs}}}{(\varDelta Z)}. \end{aligned}$$
(51)

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Wang, T., Jobredeaux, R., Pantel, M. et al. Credible autocoding of convex optimization algorithms. Optim Eng 17, 781–812 (2016). https://doi.org/10.1007/s11081-016-9320-7

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