The second volume of the Journal of Nonsmooth Analysis and Optimization (2021)
Lisa C. Hegerhorst-Schultchen ; Christian Kirches ; Marc C. Steinbach.
This work is part of an ongoing effort of comparing non-smooth optimization
problems in abs-normal form to MPCCs. We study the general abs-normal NLP with
equality and inequality constraints in relation to an equivalent MPCC
reformulation. We show that kink qualifications and MPCC constraint
qualifications of linear independence type and Mangasarian-Fromovitz type are
equivalent. Then we consider strong stationarity concepts with first and second
order optimality conditions, which again turn out to be equivalent for the two
problem classes. Throughout we also consider specific slack reformulations
suggested in [9], which preserve constraint qualifications of linear
independence type but not of Mangasarian-Fromovitz type.
Lisa C. Hegerhorst-Schultchen ; Christian Kirches ; Marc C. Steinbach.
This work continues an ongoing effort to compare non-smooth optimization
problems in abs-normal form to Mathematical Programs with Complementarity
Constraints (MPCCs). We study general Nonlinear Programs with equality and
inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their
relation to equivalent MPCC reformulations. We introduce the concepts of
Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and
MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a
specific slack reformulation suggested in [10], the relations are non-trivial.
It turns out that constraint qualifications of Abadie type are preserved. We
also prove the weaker result that equivalence of Guginard's (and Abadie's)
constraint qualifications for all branch problems hold, while the question of
GCQ preservation remains open. Finally, we introduce M-stationarity and
B-stationarity concepts for abs-normal NLPs and prove first order optimality
conditions corresponding to MPCC counterpart formulations.
Nguyen Duy Cuong ; Alexander Y. Kruger.
We propose a unifying general (i.e. not assuming the mapping to have any
particular structure) view on the theory of regularity and clarify the
relationships between the existing primal and dual quantitative sufficient and
necessary conditions including their hierarchy. We expose the typical sequence
of regularity assertions, often hidden in the proofs, and the roles of the
assumptions involved in the assertions, in particular, on the underlying space:
general metric, normed, Banach or Asplund. As a consequence, we formulate
primal and dual conditions for the stability properties of solution mappings to
inclusions
Matúš Benko.
In this paper, we study continuity and Lipschitzian properties of set-valued
mappings, focusing on inner-type conditions. We introduce new notions of inner
calmness* and, its relaxation, fuzzy inner calmness*. We show that polyhedral
maps enjoy inner calmness* and examine (fuzzy) inner calmness* of a multiplier
mapping associated with constraint systems in depth. Then we utilize these
notions to develop some new rules of generalized differential calculus, mainly
for the primal objects (e.g. tangent cones). In particular, we propose an exact
chain rule for graphical derivatives. We apply these results to compute the
derivatives of the normal cone mapping, essential e.g. for sensitivity analysis
of variational inequalities.
Matúš Benko ; Patrick Mehlitz.
Implicit variables of a mathematical program are variables which do not need
to be optimized but are used to model feasibility conditions. They frequently
appear in several different problem classes of optimization theory comprising
bilevel programming, evaluated multiobjective optimization, or nonlinear
optimization problems with slack variables. In order to deal with implicit
variables, they are often interpreted as explicit ones. Here, we first point
out that this is a light-headed approach which induces artificial locally
optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type
necessary optimality conditions which correspond to treating the implicit
variables as explicit ones on the one hand, or using them only implicitly to
model the constraints on the other. A detailed comparison of the obtained
stationarity conditions as well as the associated underlying constraint
qualifications will be provided. Overall, we proceed in a fairly general
setting relying on modern tools of variational analysis. Finally, we apply our
findings to different well-known problem classes of mathematical optimization
in order to visualize the obtained theory.
Antonio Silveti-Falls ; Cesare Molinari ; Jalal Fadili.
In this paper we propose and analyze inexact and stochastic versions of the CGALP algorithm developed in [25], which we denote ICGALP , that allow for errors in the computation of several important quantities. In particular this allows one to compute some gradients, proximal terms, and/or linear minimization oracles in an inexact fashion that facilitates the practical application of the algorithm to computationally intensive settings, e.g., in high (or possibly infinite) dimensional Hilbert spaces commonly found in machine learning problems. The algorithm is able to solve composite minimization problems involving the sum of three convex proper lower-semicontinuous functions subject to an affine constraint of the form Ax = b for some bounded linear operator A. Only one of the functions in the objective is assumed to be differentiable, the other two are assumed to have an accessible proximal operator and a linear minimization oracle. As main results, we show convergence of the Lagrangian values (so-called convergence in the Bregman sense) and asymptotic feasibility of the affine constraint as well as strong convergence of the sequence of dual variables to a solution of the dual problem, in an almost sure sense. Almost sure convergence rates are given for the Lagrangian values and the feasibility gap for the ergodic primal variables. Rates in expectation are given for the Lagrangian values and the feasibility gap subsequentially in the pointwise sense. Numerical […]
Felix Harder.
It is known in the literature that local minimizers of mathematical programs
with complementarity constraints (MPCCs) are so-called M-stationary points, if
a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds.
In this paper we present a new elementary proof for this result. Our proof is
significantly simpler than existing proofs and does not rely on deeper
technical theory such as calculus rules for limiting normal cones. A crucial
ingredient is a proof of a (to the best of our knowledge previously open)
conjecture, which was formulated in a Diploma thesis by Schinabeck.