
Semidefinite Outer Approximation of the Backward Reachable Set of Discretetime Autonomous Polynomial Systems
We approximate the backward reachable set of discretetime autonomous po...
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NeuroOptimization: Learning Objective Functions Using Neural Networks
Mathematical optimization is widely used in various research fields. Wit...
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Toptimal design for multivariate polynomial regression using semidefinite programming
We consider Toptimal experiment design problems for discriminating mult...
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Linear Encodings for Polytope Containment Problems
The polytope containment problem is deciding whether a polytope is a con...
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On the Global Optimality of ModelAgnostic MetaLearning
Modelagnostic metalearning (MAML) formulates metalearning as a bileve...
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A TwoTimescale Framework for Bilevel Optimization: Complexity Analysis and Application to ActorCritic
This paper analyzes a twotimescale stochastic algorithm for a class of ...
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Locally Feasibly Projected Sequential Quadratic Programming for Nonlinear Programming on Arbitrary Smooth Constraint Manifolds
Highdimensional nonlinear optimization problems subject to nonlinear co...
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RobusttoDynamics Optimization
A robusttodynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function f:R^n→R and a feasible set Ω⊆R^n), and (ii) a dynamical system (a map g:R^n→R^n). Its goal is to minimize f over the set S⊆Ω of initial conditions that forever remain in Ω under g. The focus of this paper is on the case where the mathematical program is a linear program and the dynamical system is either a known linear map, or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to S. Our inner approximations are optimized with respect to the objective function f and their semidefinite characterizationwhich has a semidefinite constraint of fixed sizeis obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomialtime algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case.
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