287 research outputs found
Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions
Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions
We introduce the concept of strong high-order approximate minimizers for
nonconvex optimization problems. These apply in both standard smooth and
composite non-smooth settings, and additionally allow convex or inexpensive
constraints. An adaptive regularization algorithm is then proposed to find such
approximate minimizers. Under suitable Lipschitz continuity assumptions,
whenever the feasible set is convex, it is shown that using a model of degree
, this algorithm will find a strong approximate q-th-order minimizer in at
most
evaluations of the problem's functions and their derivatives, where
is the -th order accuracy tolerance; this bound applies when
either or the problem is not composite with . For general
non-composite problems, even when the feasible set is nonconvex, the bound
becomes
evaluations. If the problem is composite, and either or the feasible
set is not convex, the bound is then evaluations. These results not only provide, to
our knowledge, the first known bound for (unconstrained or
inexpensively-constrained) composite problems for optimality orders exceeding
one, but also give the first sharp bounds for high-order strong approximate
-th order minimizers of standard (unconstrained and inexpensively
constrained) smooth problems, thereby complementing known results for weak
minimizers.Comment: 32 pages, 1 figur
- …