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Quantitative Sensitivity Bounds for Nonlinear Programming and Time-varying Optimization

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Document pages: 13 pages

Abstract: Inspired by classical sensitivity results for nonlinear optimization, wederive and discuss new quantitative bounds to characterize the solution map anddual variables of a parametrized nonlinear program. In particular, we deriveexplicit expressions for the local and global Lipschitz constants of thesolution map of non-convex or convex optimization problems, respectively. Ourresults are geared towards the study of time-varying optimization problemswhich are commonplace in various applications of online optimization, includingpower systems, robotics, signal processing and more. In this context, ourresults can be used to bound the rate of change of the optimizer. To illustratethe use of our sensitivity bounds we generalize existing arguments to quantifythe tracking performance of continuous-time, monotone running algorithms.Further, we introduce a new continuous-time running algorithm for time-varyingconstrained optimization which we model as a so-called perturbed sweepingprocess. For this discontinuous scheme, we establish an explicit bound on theasymptotic solution tracking for a class of convex problems.

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