246 research outputs found
Strong Stationarity Conditions for Optimal Control of Hybrid Systems
We present necessary and sufficient optimality conditions for finite time
optimal control problems for a class of hybrid systems described by linear
complementarity models. Although these optimal control problems are difficult
in general due to the presence of complementarity constraints, we provide a set
of structural assumptions ensuring that the tangent cone of the constraints
possesses geometric regularity properties. These imply that the classical
Karush-Kuhn-Tucker conditions of nonlinear programming theory are both
necessary and sufficient for local optimality, which is not the case for
general mathematical programs with complementarity constraints. We also present
sufficient conditions for global optimality.
We proceed to show that the dynamics of every continuous piecewise affine
system can be written as the optimizer of a mathematical program which results
in a linear complementarity model satisfying our structural assumptions. Hence,
our stationarity results apply to a large class of hybrid systems with
piecewise affine dynamics. We present simulation results showing the
substantial benefits possible from using a nonlinear programming approach to
the optimal control problem with complementarity constraints instead of a more
traditional mixed-integer formulation.Comment: 30 pages, 4 figure
Global error bounds for convex conic problems
In this paper Lipschitzian type error bounds are derived for general convex conic problems under various regularity conditions.
Specifically, it is shown that if the recession directions satisfy Slater's condition
then a global Lipschitzian type error bound holds. Alternatively, if the feasible region is bounded, then the ordinary Slater condition guarantees a global Lipschitzian type error bound. These can be considered as generalizations of previously known results for inequality systems. Moreover, some of the results are
also generalized to the intersection of multiple cones. Under Slater's condition alone, a global Lipschitzian type error bound may not hold. However, it is shown that such an error bound holds for a specific region.
For linear systems we show that the constant involved in Hoffman's error bound can be estimated by the so-called condition number for linear programming
Model Consistency for Learning with Mirror-Stratifiable Regularizers
Low-complexity non-smooth convex regularizers are routinely used to impose
some structure (such as sparsity or low-rank) on the coefficients for linear
predictors in supervised learning. Model consistency consists then in selecting
the correct structure (for instance support or rank) by regularized empirical
risk minimization.
It is known that model consistency holds under appropriate non-degeneracy
conditions. However such conditions typically fail for highly correlated
designs and it is observed that regularization methods tend to select larger
models.
In this work, we provide the theoretical underpinning of this behavior using
the notion of mirror-stratifiable regularizers. This class of regularizers
encompasses the most well-known in the literature, including the or
trace norms. It brings into play a pair of primal-dual models, which in turn
allows one to locate the structure of the solution using a specific dual
certificate.
We also show how this analysis is applicable to optimal solutions of the
learning problem, and also to the iterates computed by a certain class of
stochastic proximal-gradient algorithms.Comment: 14 pages, 4 figure
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