In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on large-scale semidefinite programs, however, has limited the scale of problems to which it can be applied. In this paper, we introduce diagonally dominant sum of squares (DSOS) and scaled diagonally dominant sum of squares (SDSOS) optimization as linear programming and second-order cone programming-based alternatives to sum of squares optimization that allow one to trade off computation time with solution quality. These are optimization problems over certain subsets of sum of squares polynomials (or equivalently subsets of positive semidefinite matrices), which can be of interest in general applications of semidefinite programming where scalability is a limitation. We show that some basic theorems from SOS optimization which rely on results from real algebraic geometry are still valid for DSOS and SDSOS optimization. Furthermore, we show with numerical experiments from diverse application areas-polynomial optimization, statistics and machine learning, derivative pricing, and control theory-that with reasonable trade-offs in accuracy, we can handle problems at scales that are currently significantly beyond the reach of traditional sum of squares approaches. Finally, we provide a review of recent techniques that bridge the gap between our DSOS/SDSOS approach and the SOS approach at the expense of additional running time. The supplementary material to the paper introduces an accompanying MATLAB package for DSOS and SDSOS optimization.

AimsCoronary plaque characteristics are associated with ischaemia. Differences in plaque volumes and composition may explain the discordance between coronary stenosis severity and ischaemia. We evaluated the association between coronary stenosis severity, plaque characteristics, coronary computed tomography angiography (CTA)-derived fractional flow reserve (FFRCT), and lesion-specific ischaemia identified by FFR in a substudy of the NXT trial (Analysis of Coronary Blood Flow Using CT Angiography: Next Steps).Methods and resultsCoronary CTA stenosis, plaque volumes, FFRCT, and FFR were assessed in 484 vessels from 254 patients. Stenosis >50% was considered obstructive. Plaque volumes (non-calcified plaque [NCP], low-density NCP [LD-NCP], and calcified plaque [CP]) were quantified using semi-automated software. Optimal thresholds of quantitative plaque variables were defined by area under the receiver-operating characteristics curve (AUC) analysis. Ischaemia was defined by FFR or FFRCT ≤0.80. Plaque volumes were inversely related to FFR irrespective of stenosis severity. Relative risk (95% confidence interval) for prediction of ischaemia for stenosis >50%, NCP ≥185 mm3, LD-NCP ≥30 mm3, CP ≥9 mm3, and FFRCT ≤0.80 were 5.0 (3.0–8.3), 3.7 (2.4–5.6), 4.6 (2.9–7.4), 1.4 (1.0–2.0), and 13.6 (8.4–21.9), respectively. Low-density NCP predicted ischaemia independent of other plaque characteristics. Low-density NCP and FFRCT yielded diagnostic improvement over stenosis assessment with AUCs increasing from 0.71 by stenosis >50% to 0.79 and 0.90 when adding LD-NCP ≥30 mm3 and LD-NCP ≥30 mm3 + FFRCT ≤0.80, respectively.ConclusionStenosis severity, plaque characteristics, and FFRCT predict lesion-specific ischaemia. Plaque assessment and FFRCT provide improved discrimination of ischaemia compared with stenosis assessment alone.

Abstract-Motivated by the need for formal guarantees on the stability and safety of controllers for challenging robot control tasks, we present a control design procedure that explicitly seeks to maximize the size of an invariant "funnel" that leads to a predefined goal set. Our certificates of invariance are given in terms of sums of squares proofs of a set of appropriately defined Lyapunov inequalities. These certificates, together with our proposed polynomial controllers, can be efficiently obtained via semidefinite optimization. Our approach can handle time-varying dynamics resulting from tracking a given trajectory, input saturations (e.g. torque limits), and can be extended to deal with uncertainty in the dynamics and state. The resulting controllers can be used by space-filling feedback motion planning algorithms to fill up the space with significantly fewer trajectories. We demonstrate our approach on a severely torque limited underactuated double pendulum (Acrobot) and provide extensive simulation and hardware validation.

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