Giovanni Fantuzzi

Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization

We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit c...

Fast ADMM for semidefinite programs with chordal sparsity

Many problems in control theory can be formulated as semidefinite programs (SDPs). 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). We show that chordal decomposition can be applied to either the primal or the dual standard form of a sparse SDP, resulting in scaled versions of ADMM algorithms with the same computational cost. Each iteration of our algorithms consists of a projection on the product of small positive semidefinite cones, followed by a projection on an affine set, both of which can be carried out efficiently. Our techniques are implemented in CDCS, an open source add-on to MATLAB. Numerical experiments on large-scale sparse problems in SDPLIB and random SDPs with block-arrow sparse patterns show speedups compared to some common state-of-the-art software packages.

Fast ADMM for homogeneous self-dual embedding of sparse SDPs * *Y. Zheng and G. Fantuzzi contributed equally to this work. Y. Zheng is supported by the Clarendon Scholarship and the Jason Hu Scholarship.

Abstract We propose an efficient first-order method, based on the alternating direction method of multipliers (ADMM), to solve the homogeneous self-dual embedding problem for a primal-dual pair of semidefinite programs (SDPs) with chordal sparsity. Using a series of block eliminations, the per-iteration cost of our method is the same as applying a splitting method to the primal or dual alone. Moreover, our approach is more efficient than other first-order methods for generic sparse conic programs since we work with smaller semidefinite cones. In contrast to previous first-order methods that exploit chordal sparsity, our algorithm returns both primal and dual solutions when available, and a certificate of infeasibility otherwise. Our techniques are implemented in the open-source MATLAB solver CDCS. Numerical experiments on three sets of benchmark problems from the library SDPLIB show speed-ups compared to some common state-of-the-art software packages.

Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints

We introduce a new technique to optimize a linear cost function subject to an affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional affine in the optimization variables is nonnegative over a space of functions defined by homogeneous boundary conditions. Such problems arise in control and stability or input-to-state/output analysis of systems governed by partial differential equations (PDEs), particularly fluid dynamical systems. We derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that a convergent sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). We also obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can be computed with SDPs. We present QUINOPT, an open-source add-on to YALMIP to aid the formulation and solution of our SDPs, and demonstrate our techniques on problems arising from the stability analysis of PDEs.

Optimal bounds with semidefinite programming: An application to stress driven shear flows

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite-dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite-dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress-driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.

Exploiting Sparsity in the Coefficient Matching Conditions in Sum-of-Squares Programming Using ADMM

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. Each iteration of our algorithm requires one projection onto the positive semidefinite cone and the solution of multiple quadratic programs with closed-form solutions free of any matrix inversion. Our techniques are implemented in the open-source MATLAB solver SOSADMM. Numerical experiments on SOS problems arising from unconstrained polynomial minimization and from Lyapunov stability analysis for polynomial systems show speed-ups compared to the interior-point solver SeDuMi, and the first-order solver CDCS.

Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh–Benard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number

Semidefinite relaxation of a class of quadratic integral inequalities

\mathit{Nu}

Construction of an Optimal Background Profile for the Kuramoto-Sivashinsky Equation using Semidefinite Programming

is bounded by

Chordal decomposition in operator-splitting methods for sparse semidefinite programs

\mathit{Nu}\leqslant 0.5999\mathit{R}^{1/3}