Exploiting sparsity for neural network verification by Newton, M, Papachristodoulou, A

“…There has been significant progress to improve the efficiency and the accuracy of these methods. We investigate the sparsity that arises in a recently proposed semi-definite programming framework to verify a fully connected feed-forward neural network. …”

Chordal sparsity, decomposing SDPs and the Lyapunov equation by Mason, R, Papachristodoulou, A

“…For large LMIs it is important to exploit structure and sparsity within the problem in order to solve the associated Semidefinite Programs efficiently. …”

Fast ADMM for semidefinite programs with chordal sparsity by Zheng, Y, Fantuzzi, G, Papachristodoulou, A, Goulart, P, Wynn, A

“…For large-scale SDPs, it is important to exploit the inherent sparsity to improve scalability. This paper develops efficient first-order methods to solve SDPs with chordal sparsity based on the alternating direction method of multipliers (ADMM). …”

Chordal sparsity in control and optimization of large-scale systems by Zheng, Y

“…</p> <p>The first part of this thesis proposes a new conversion framework for large-scale SDPs characterized by chordal sparsity. This framework is analogous to standard conversion techniques for interior-point methods, but is more suitable for the application of first-order methods. …”

A chordal sparsity approach to scalable linear and nonlinear systems analysis by Mason, R

“…<p>In this thesis we investigate how the properties of chordal graphs can be used to exploit sparsity in several optimisation problems that arise in control theory. …”

Exploiting sparsity in the coefficient matching conditions in sum-of-squares programming using ADMM by Zheng, Y, Fantuzzi, G, Papachristodoulou, A

“…This letter introduces an efficient first-order method based on the alternating direction method of multipliers (ADMM) to solve semidefinite programs arising from sum-of-squares (SOS) programming. We exploit the sparsity of the coefficient matching conditions when SOS programs are formulated in the usual monomial basis to reduce the computational cost of the ADMM algorithm. …”

Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations by Ahmadi, AA, Hall, G, Papachristodoulou, A, Saunderson, J, Zheng, Y

“…The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. …”

Scalable design of structured controllers using chordal decomposition by Zheng, Y, Mason, R, Papachristodoulou, A

“…We first extend the chordal decomposition theorem for positive semidefinite matrices to the case of matrices with block-chordal sparsity. Then, a block-diagonal Lyapunov matrix assumption is used to convert the design of structured feedback gains into a convex problem, which inherits the sparsity pattern of the original problem. …”

Decomposition and completion of sum-of-squares matrices by Zheng, Y, Fantuzzi, G, Papachristodoulou, A

“…We show that a subset of sparse SOS matrices with chordal sparsity patterns can be equivalently decomposed into a sum of multiple SOS matrices that are nonzero only on a principal submatrix. …”

Distributed design for decentralized control using chordal decomposition and ADMM by Zheng, Y, Kamgarpour, M, Sootla, A, Papachristodoulou, A

“…We propose a distributed design method for decentralized control by exploiting the underlying sparsity properties of the problem. Our method is based on the chordal decomposition of sparse block matrices and the alternating direction method of multipliers (ADMM). …”

Chordal decomposition in rank minimized semidefinite programs with applications to subspace clustering by Miller, J, Zheng, Y, Roig-Solvas, B, Sznaier, M, Papachristodoulou, A

“…Decomposition methods based on chordal sparsity have already been applied to speed up the solution of sparse SDPs, but methods for dealing with rank constraints are underdeveloped. …”

Analysis of robust neural networks for control by Newton, M

“…We reformulate and exploit the sparsity in the optimisation problem, showing a significant speed-up in computation. …”

Fast ADMM for homogeneous self-dual embeddings of sparse SDPs by Zheng, Y, Fantuzzi, G, Papachristodoulou, A, Goulart, P, Wynn, A

“…In contrast to previous first-order methods that exploit chordal sparsity, our algorithm returns both primal and dual solutions when available, and it provides a certificate of infeasibility otherwise. …”

SOSTOOLS: Control applications and new developments by Prajna, S, Papachristodoulou, A, Seiler, P, Parrilo, P

“…The nonlinear stability analysis, parametric robustness analysis, safety verification, and nonlinear controller synthesis was also considered in the control applications. A sparsity structure was used while considering infinitely constrained linear matrix inequalities (LMI).…”

Structured sum of squares for networked systems analysis. by Hancock, E, Papachristodoulou, A

“…By taking the structure of the network into account, we limit the size and number of decision variables in the LMI representation of the Sum of Squares, which improves the scalability of the technique for networked systems beyond taking advantage of symmetry and sparsity. We apply the technique to test non-negativity of fourth order structured polynomials in many variables and show that for these problems the technique has improved scalability over existing Sum of Squares techniques. © 2011 IEEE.…”

A Decomposition Technique for Nonlinear Dynamical System Analysis. by Anderson, J, Papachristodoulou, A

“…Further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem Lyapunov functions. © 2011 IEEE.…”

Dynamical system decomposition for efficient, sparse analysis. by Anderson, J, Papachristodoulou, A

“…A new method to reduce the number of decision variables in SOS programmes by maximizing the sparsity of the coefficients in the subsystem certificates is also presented. ©2010 IEEE.…”

A chordal decomposition approach to scalable design of structured feedback gains over directed graphs by Zheng, Y, Mason, R, Papachristodoulou, A

“…This paper considers the problem of designing static feedback gains subject to a priori structural constraints, which is in general a non-convex problem. By exploiting the sparsity properties of the problem, and using chordal decomposition, a scalable algorithm is proposed to compute structured stabilizing feedback gains for large-scale systems over directed graphs. …”

Decomposed structured subsets for semidefinite and sum-of-squares optimization by Miller, J, Zheng, Y, Sznaier, M, Papachristodoulou, A

“…Meanwhile, any underlying sparsity or symmetry structure may be leveraged to form an equivalent SDP with smaller positive semidefinite constraints. …”

Neural network verification using polynomial optimisation by Newton, M, Papachristodoulou, A

“…This method can be extended to more activation functions, and combined with recent sparsity-exploiting methods can result in a computationally acceptable method for verifying neural networks.…”