First-order conditions for isolated locally optimal solutions
Journal of Optimization Theory and Applications
SIAM Journal on Control and Optimization
Nonsmooth analysis and control theory
Nonsmooth analysis and control theory
Robust decision problems in engineering: a linear matrix inequality approach
Advances in linear matrix inequality methods in control
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
Optimization and Pseudospectra, with Applications to Robust Stability
SIAM Journal on Matrix Analysis and Applications
Solution Continuity in Monotone Affine Variational Inequalities
SIAM Journal on Optimization
Variational Analysis of Pseudospectra
SIAM Journal on Optimization
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To minimize or upper-bound the value of a function “robustly,” we might instead minimize or upper-bound the “$\epsilon$-robust regularization,” defined as the map from a point to the maximum value of the function within an $\epsilon$-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius of a matrix leads to pseudospectral computations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small $\epsilon0$, even if the original function is non-Lipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such functions have graphs that are finite unions of sets defined by finitely many polynomial inequalities, and are commonly encountered in applications.