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The combinatorial integral approximation (CIA) is a solution technique for integer optimal control problems. In order to regularize the solutions produced by CIA, one can minimize switching costs in one of its algorithmic steps. This leads to combinatorial optimization problems, which are called switching cost aware rounding problems (SCARP). They can be solved efficiently on one-dimensional domains but no efficient solution algorithms have been found so far for multi-dimensional domains. The CIA problem formulation depends on a discretization grid. We propose to reduce the number of variables and thus improve the computational tractability of SCARP by means of a non-uniform grid refinement strategy. We prove that the grid refinement preserves the approximation properties of the combinatorial integral approximation. Computational results are offered to show that the proposed approach is able to achieve, within a prescribed time limit, smaller duality gaps that does the uniform approach. For several large instances, a dual bound could only be obtained through adaptivity.

Gradient-based methods have been highly successful for solving a variety of both unconstrained and constrained nonlinear optimization problems. In real-world applications, such as optimal control or machine learning, the necessary function and derivative information may be corrupted by noise, however. Sun and Nocedal have recently proposed a remedy for smooth unconstrained problems by means of a stabilization of the acceptance criterion for computed iterates, which leads to convergence of the iterates of a trust-region method to a region of criticality, Sun and Nocedal (2022). We extend their analysis to the successive linear programming algorithm, Byrd et al. (2023a,2023b), for unconstrained optimization problems with objectives that can be characterized as the composition of a polyhedral function with a smooth function, where the latter and its gradient may be corrupted by noise. This gives the flexibility to cover, for example, (sub)problems arising image reconstruction or constrained optimization algorithms. We provide computational examples that illustrate the findings and point to possible strategies for practical determination of the stabilization parameter that balances the size of the critical region with a relaxation of the acceptance criterion (or descent property) of the algorithm.

Binary trust-region steepest descent (BTR) and combinatorial integral approximation (CIA) are two recently investigated approaches for the solution of optimization problems with distributed binary-/discrete-valued variables (control functions). We show improved convergence results for BTR by imposing a compactness assumption that is similar to the convergence theory of CIA. As a corollary we conclude that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly-$^*$ to stationary points of the latter. We provide computational results that validate our findings. In addition, we observe a regularizing effect of BTR, which we explore by means of a hybridization of CIA and BTR.

Statistical matching methods are widely used in the social and health sciences to estimate causal effects using observational data. Often the objective is to find comparable groups with similar covariate distributions in a dataset, with the aim to reduce bias in a random experiment. We aim to develop a foundation for deterministic methods which provide results with low bias, while retaining interpretability. The proposed method matches on the covariates and calculates all possible maximal exact matchesfor a given dataset without adding numerical errors. Notable advantages of our method over existing matching algorithms are that all available information for exact matches is used, no additional bias is introduced, it can be combined with other matching methods for inexact matching to reduce pruning and that the result is calculated in a fast and deterministic way. For a given dataset the result is therefore provably unique for exact matches in the mathematical sense. We provide proofs, instructions for implementation as well as a numerical example calculated for comparison on a complete survey.

We propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to obtain an estimate of the active set. The algorithm enforces descent on the objective function to promote global convergence to B-stationary points. We provide a convergence analysis and preliminary numerical results on a range of test problems. We also study the effect of fixing the active constraints in a bound-constrained quadratic program that can be solved on each iteration in order to obtain fast convergence., Comment: 33 pages

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.

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.

Propensity score matching (PSM) is the de-facto standard for estimating causal effects in observational studies. We show that PSM and its implementations are susceptible to several major drawbacks and illustrate these findings using a case study with $17,427$ patients. We derive four formal properties an optimal statistical matching algorithm should meet, and propose Deterministic Balancing Score exact Matching (DBSeM) which meets the aforementioned properties for an exact matching. Furthermore, we investigate one of the main problems of PSM, that is that common PSM results in one valid set of matched pairs or a bootstrapped PSM in a selection of possible valid sets of matched pairs. For exact matchings we provide the mathematical proof, that DBSeM, as a result, delivers the expected value of all valid sets of matched pairs for the investigated dataset., Comment: 25 pages, 3 tables

We describe trlib, a library that implements a variant of Gould's Generalized Lanczos method (Gould et al. in SIAM J. Opt. 9(2), 504-525, 1999) for solving the trust region problem. Our implementation has several distinct features that set it apart from preexisting ones. We implement both conjugate gradient (CG) and Lanczos iterations for assembly of Krylov subspaces. A vector- and matrix-free reverse communication interface allows the use of most general data structures, such as those arising after discretization of function space problems. The hard case of the trust region problem frequently arises in sequential methods for nonlinear optimization. In this implementation, we made an effort to fully address the hard case in an exact way by considering all invariant Krylov subspaces. We investigate the numerical performance of trlib on the full subset of unconstrained problems of the CUTEst benchmark set. In addition to this, interfacing the PDE discretization toolkit FEniCS with trlib using the vector-free reverse communication interface is demonstrated for a family of PDE-constrained control trust region problems adapted from the OPTPDE collection.