The inaugural volume of the Journal of Nonsmooth Analysis and Optimization (2020)
Hannes Meinlschmidt ; Christian Meyer ; Stephan Walther.
The paper is concerned with an optimal control problem governed by a state
equation in form of a generalized abstract operator differential equation
involving a maximal monotone operator. The state equation is uniquely solvable,
but the associated solution operator is in general not
Gâteaux-differentiable. In order to derive optimality conditions, we
therefore regularize the state equation and its solution operator,
respectively, by means of a (smoothed) Yosida approximation. We show
convergence of global minimizers for regularization parameter tending to zero
and derive necessary and sufficient optimality conditions for the regularized
problems. The paper ends with an application of the abstract theory to optimal
control of homogenized quasi-static elastoplasticity.
Kamil A. Khan ; Yingwei Yuan.
For any scalar-valued bivariate function that is locally Lipschitz continuous
and directionally differentiable, it is shown that a subgradient may always be
constructed from the function's directional derivatives in the four compass
directions, arranged in a so-called "compass difference". When the original
function is nonconvex, the obtained subgradient is an element of Clarke's
generalized gradient, but the result appears to be novel even for convex
functions. The function is not required to be represented in any particular
form, and no further assumptions are required, though the result is
strengthened when the function is additionally L-smooth in the sense of
Nesterov. For certain optimal-value functions and certain parametric solutions
of differential equation systems, these new results appear to provide the only
known way to compute a subgradient. These results also imply that centered
finite differences will converge to a subgradient for bivariate nonsmooth
functions. As a dual result, we find that any compact convex set in two
dimensions contains the midpoint of its interval hull. Examples are included
for illustration, and it is demonstrated that these results do not extend
directly to functions of more than two variables or sets in higher dimensions.
Patrick Mehlitz.
Based on the tools of limiting variational analysis, we derive a sequential
necessary optimality condition for nonsmooth mathematical programs which holds
without any additional assumptions. In order to ensure that stationary points
in this new sense are already Mordukhovich-stationary, the presence of a
constraint qualification which we call AM-regularity is necessary. We
investigate the relationship between AM-regularity and other constraint
qualifications from nonsmooth optimization like metric (sub-)regularity of the
underlying feasibility mapping. Our findings are applied to optimization
problems with geometric and, particularly, disjunctive constraints. This way,
it is shown that AM-regularity recovers recently introduced
cone-continuity-type constraint qualifications, sometimes referred to as
AKKT-regularity, from standard nonlinear and complementarity-constrained
optimization. Finally, we discuss some consequences of AM-regularity for the
limiting variational calculus.
Kevin Sturm.
In this paper we study the right differentiability of a parametric infimum
function over a parametric set defined by equality constraints. We present a
new theorem with sufficient conditions for the right differentiability with
respect to the parameter. Target applications are nonconvex objective functions
with equality constraints arising in optimal control and shape optimisation.
The theorem makes use of the averaged adjoint approach in conjunction with the
variational approach of Kunisch, Ito and Peichl. We provide two examples of our
abstract result: (a) a shape optimisation problem involving a semilinear
partial differential equation which exhibits infinitely many solutions, (b) a
finite dimensional quadratic function subject to a nonlinear equation.