arXiv subjects

Sebastian Lämmel

Publications and source records attributed to Sebastian Lämmel.

8 recordsLinked to original sources

arXiv2026

Solving mathematical programs with complementarity constraints by disjunctive regularizations

We propose a new disjunctive regularization for mathematical programs with complementarity constraints (MPCC). Its feasible set coincides with that of the Kanzow-Schwartz regularization. However, their functional descriptions differ considerably. For the disjunctive regularization, the logical operator OR and equivalent max-type constraints are used. Unlike the Kanzow-Schwartz, the disjunctive regularization satisfies the tailored linear independence constraint qualification if the original MPCC does. More than that, the favorable convergence properties - known to hold for the Kanzow-Schwartz regularization - remain valid for the disjunctive regularization as well. In particular, no second order necessary conditions are required to guarantee convergence towards S-stationary points of MPCC. Additionally, we keep track of the topological type of approximating and limiting nondegenerate C-stationary points in terms of their C-indices. Quadratic and biactive parts of the C-indices are shown to generically correspond to each other while regularizing. This is a new phenomenon as compared to the Scholtes or sign-type regularizations studied before. Numerical experiments illustrate that the proposed disjunctive regularization clearly outperforms the Kanzow-Schwartz regularization. Its numerical performance is even better than that of the Scholtes regularization if solving MPCCs with high accuracy.

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arXiv2025

On stability of the Scholtes regularization for mathematical programs with complementarity constraints

For mathematical programs with complementarity constraints (MPCC), we study the stability properties of their Scholtes regularization. Our goal is to relate nondegenerate C-stationary points of MPCC with nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization up to their topological type. As it is standard in the framework of Morse theory, the topological types are captured by the C-index and the quadratic index, respectively. It turns out that a change of the topological type for the approximating Karush-Kuhn-Tucker points of the Scholtes regularization and their limiting C-stationary point is possible. In particular, a minimizer of MPCC with zero C-index might be approximated by saddle points of the Scholtes regularization with nonzero quadratic index. In order to bypass this index shift phenomenon, an additional generic condition for nondegenerate C-stationary points of MPCC is identified. It says that non-biactive multipliers under consideration should not vanish. Then, we uniquely trace nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization and successively maintain the topological type of their limiting C-stationary point. The main technical issue here is to relate the first-order information of the defining functions, which enters the biactive part of the C-index, with the second-order information, which enters the quadratic index of the Karush-Kuhn-Tucker points.

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arXiv2022

Global aspects of the continuous reformulation for cardinality-constrained optimization problems

The main goal of this paper is to relate the topologically relevant stationary points of a cardinality-constrained optimization problem and its continuous reformulation up to their type. For that, we focus on the nondegenerate M- and T-stationary points, respectively. Their so-called M- and T-indices, which uniquely determine the global and local structure of optimization problems under consideration in algebraic terms, are traced. As novelty, we suggest to regularize the continuous reformulation for this purpose. The main consequence of our analysis is that the number of saddle points of the regularized continuous reformulation grows exponentially as compared to that of the initial cardinality-constrained optimization problem. Additionally, we obtain the Morse theory for the regularized continuous reformulation by using the corresponding results on mathematical programs with orthogonality type constraints.

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arXiv2021

Optimality conditions for mathematical programs with orthogonality type constraints

We consider the class of mathematical programs with orthogonality type constraints (MPOC). Orthogonality type constraints appear by reformulating the sparsity constraint via auxiliary binary variables and relaxing them afterwards. For MPOC a necessary optimality condition in terms of T-stationarity is stated. The justification of T-stationarity is threefold. First, it allows to capture the global structure of MPOC in terms of Morse theory, i. e. deformation and cell-attachment results are established. For that, nondegeneracy for the T-stationary points is introduced and shown to hold at a generic MPOC. Second, we prove that Karush-Kuhn-Tucker points of the Scholtes-type regularization converge to T-stationary points of MPOC. This is done under the MPOC-tailored linear independence constraint qualification (LICQ), which turns out to be a generic property too. Third, we show that T-stationarity applied to the relaxation of sparsity constrained nonlinear optimization (SCNO) naturally leads to its M-stationary points. Moreover, we argue that all T-stationary points of this relaxation become degenerate.

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arXiv2021

Cardinality-constrained optimization problems in general position and beyond

We study cardinality-constrained optimization problems (CCOP) in general position, i. e. those optimization-related properties that are fulfilled for a dense and open subset of their defining functions. We show that the well-known cardinality-constrained linear independence constraint qualification (CC-LICQ) is generic in this sense. For M-stationary points we define nondegeneracy and show that it is a generic property too. In particular, the sparsity constraint turns out to be active at all minimizers of a generic CCOP. Moreover, we describe the global structure of CCOP in the sense of Morse theory, emphasizing the strength of the generic approach. Here, we prove that multiple cells need to be attached, each of dimension coinciding with the proposed M-index of nondegenerate M-stationary points. Beyond this generic viewpoint, we study singularities of CCOP. For that, the relation between nondegeneracy and strong stability in the sense of Kojima (1980) is examined. We show that nondegeneracy implies the latter, while the reverse implication is in general not true. To fill the gap, we fully characterize the strong stability of M-stationary points under CC-LICQ by first- and second-order information of CCOP defining functions. Finally, we compare nondegeneracy and strong stability of M-stationary points with second-order sufficient conditions recently introduced in the literature.

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arXiv2020

On nondegenerate M-stationary points for sparsity constrained nonlinear optimization

We study sparsity constrained nonlinear optimization (SCNO) from a topological point of view. Special focus will be on M-stationary points from Burdakov et al. (2016). We introduce nondegenerate M-stationary points and define their M-index. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic SCNO. Some relations to other stationarity concepts, such as S-stationarity, basic feasibility, and CW-minimality, are discussed in detail. By doing so, the issues of instability and degeneracy of points due to different stationarity concepts are highlighted. The concept of M-stationarity allows to adequately describe the global structure of SCNO along the lines of Morse theory. For that, we study topological changes of lower level sets while passing an M-stationary point. As novelty for SCNO, multiple cells of dimension equal to the M-index are needed to be attached. This intriguing fact is in strong contrast with other optimization problems considered before, where just one cell suffices. As a consequence, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one. The appearance of such saddle points cannot be thus neglected from the perspective of global optimization. Due to the multiplicity phenomenon in cell-attachment, a saddle point may lead to more than two different local minimizers. We conclude that the relatively involved structure of saddle points is the source of well-known difficulty if solving SCNO to global optimality.

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arXiv2020

Critical point theory for sparse recovery

We study the problem of sparse recovery in the context of compressed sensing. This is to minimize the sensing error of linear measurements by sparse vectors with at most $s$ non-zero entries. We develop the so-called critical point theory for sparse recovery. This is done by introducing nondegenerate M-stationary points which adequately describe the global structure of this nonconvex optimization problem. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic sparse recovery problem. Additionally, the equivalence of strong stability and nondegeneracy for M-stationary points is shown. We claim that the appearance of saddle points - these are M-stationary points with exactly $s-1$ non-zero entries - cannot be neglected. For this purpose we derive a so-called Morse relation, which gives a lower bound on the number of saddle points in terms of the number of local minimizers. The relatively involved structure of saddle points can be seen as a source of well-known difficulty by solving the problem of sparse recovery to global optimality.

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arXiv2019

Shannon's comparison of channels characterized by optimal decision making

According to Blackwell's Theorem it is equivalent to compare channels by either a garbling order or optimal decision making. This equivalence does not hold anymore if also allowing pre-garbling, i. e. for the so-called Shannon-order (see Rauh et al., 2017). We show that the equivalence Fails in general even if the set of decision makers is reduced. This is overcome by the introduction of convexified Shannon-usefulness as a preference relation of decision makers over channels. We prove that convexified Shannon-order and convexified Shannon-usefulness are equivalent.

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