Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper

An Analysis of Zero Set and Global Error Bound Properties of a Piecewise Affine Function Via Its Recession Function

For a piecewise affine function f : R n ! R m , the recession function is defined by f 1 (x) := lim !1 f(x) : In this paper, we study the zero set and error bound properties of f via f 1 . We show, for example, that f has a zero when f 1 has a unique zero (at the origin) with a nonvanishing index. We also characterize the global error bound property of a piecewise affine function in terms of the recession cones of the zero sets of the function and its recession function. Key words. piecewise affine function, recession function, error bounds, affine variational inequality, linear complementarity problem. AMS Subject Classification. 90C30, 90C33, 49J40, 54C60 Research supported by the National Science Foundation Grant CCR-9307685 1 Introduction Consider a piecewise affine function f : R n ! R m . This means that that f is continuous and R n admits a polyhedral subdivision f\Omega 1 ;\Omega 2 ; : : : ;\Omega L g such that f(x) = A j x + a j on\Omega j (j ..