192 research outputs found
A convex analysis approach to optimal controls with switching structure for partial differential equations
Optimal control problems involving hybrid binary-continuous control costs are
challenging due to their lack of convexity and weak lower semicontinuity.
Replacing such costs with their convex relaxation leads to a primal-dual
optimality system that allows an explicit pointwise characterization and whose
Moreau-Yosida regularization is amenable to a semismooth Newton method in
function space. This approach is especially suited for computing switching
controls for partial differential equations. In this case, the optimality gap
between the original functional and its relaxation can be estimated and shown
to be zero for controls with switching structure. Numerical examples illustrate
the effectiveness of this approach
Infinite horizon sparse optimal control
A class of infinite horizon optimal control problems involving -type
cost functionals with is discussed. The existence of optimal
controls is studied for both the convex case with and the nonconvex case
with , and the sparsity structure of the optimal controls promoted by
the -type penalties is analyzed. A dynamic programming approach is
proposed to numerically approximate the corresponding sparse optimal
controllers
Sparse and switching infinite horizon optimal controls with mixed-norm penalizations
© 2020 EDP Sciences, SMAI. A class of infinite horizon optimal control problems involving mixed quasi-norms of Lp-type cost functionals for the controls is discussed. These functionals enhance sparsity and switching properties of the optimal controls. The existence of optimal controls and their structural properties are analyzed on the basis of first order optimality conditions. A dynamic programming approach is used for numerical realization
On the monotone and primal-dual active set schemes for -type problems,
Nonsmooth nonconvex optimization problems involving the quasi-norm,
, of a linear map are considered. A monotonically convergent
scheme for a regularized version of the original problem is developed and
necessary optimality conditions for the original problem in the form of a
complementary system amenable for computation are given. Then an algorithm for
solving the above mentioned necessary optimality conditions is proposed. It is
based on a combination of the monotone scheme and a primal-dual active set
strategy. The performance of the two algorithms is studied by means of a series
of numerical tests in different cases, including optimal control problems,
fracture mechanics and microscopy image reconstruction
Continuous dependence estimates and homogenization of quasi-monotone systems of fully nonlinear second order parabolic equations
Aim of this paper is to extend the continuous dependence estimates proved in
\cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic
equations. As by-product of these estimates, we get an H\"older estimate for
bounded solutions of systems and a rate of convergence estimate for the
vanishing viscosity approximation. In the second part of the paper we employ
similar techniques to study the periodic homogenization of quasi-monotone
systems of fully nonlinear second-order uniformly parabolic equations. Finally,
some examples are discussed
Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates
We consider homogenization for weakly coupled systems of Hamilton--Jacobi
equations with fast switching rates. The fast switching rate terms force the
solutions converge to the same limit, which is a solution of the effective
equation. We discover the appearance of the initial layers, which appear
naturally when we consider the systems with different initial data and analyze
them rigorously. In particular, we obtain matched asymptotic solutions of the
systems and rate of convergence. We also investigate properties of the
effective Hamiltonian of weakly coupled systems and show some examples which do
not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness
Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established. © The authors
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