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When dealing with general Lipschitzian optimization problems, there are many problem classes where standard constraint qualifications fail at local minimizers. In contrast to a constraint qualification, a problem qualification does not only rely on the constraints but also on the objective function to guarantee that a local minimizer is a Karush-Kuhn-Tucker (KKT) point. For example, calmness in the sense of Clarke is a problem qualification. In this article, we introduce the Subset Mangasarian-Fromovitz Condition (subMFC). This new problem qualification is based on a nonsmooth version of the approximate KKT conditions, which hold at every local minimizer without further assumptions. A comparison with existing constraint qualifications and problem qualifications for the given problem class reveals that subMFC is strictly weaker than quasinormality and can hold even if the local error bound condition, the cone-continuity property, Guignard's constraint qualification and calmness in the sense of Clarke are violated. Further, we emphasize the power of the new problem qualification within the context of bilevel optimization. More precisely, under mild assumptions on the problem data, we suggest a version of subMFC that is tailored to the lower-level value function reformulation. It turns out that this new condition can be satisfied even if the widely used partial calmness condition does not hold.

A highly relevant problem of modern finance is the design of Value-at-Risk (VaR) optimal portfolios. Due to contemporary financial regulations, banks and other financial institutions are tied to use the risk measure to control their credit, market and operational risks. For a portfolio with a discrete return distribution and finitely many scenarios, a Difference of Convex (DC) functions representation of the VaR can be derived. Wozabal (2012) showed that this yields a solution to a VaR constrained Markowitz style portfolio selection problem using the Difference of Convex Functions Algorithm (DCA). A recent algorithmic extension is the so-called Boosted Difference of Convex Functions Algorithm (BDCA) which accelerates the convergence due to an additional line search step. It has been shown that the BDCA converges linearly for solving non-smooth quadratic problems with linear inequality constraints. In this paper, we prove that the linear rate of convergence is also guaranteed for a piecewise linear objective function with linear equality and inequality constraints using the Kurdyka-{\L}ojasiewicz property. An extended case study under consideration of best practices for comparing optimization algorithms demonstrates the superiority of the BDCA over the DCA for real-world financial market data. We are able to show that the results of the BDCA are significantly closer to the efficient frontier compared to the DCA. Due to the open availability of all data sets and code, this paper further provides a practical guide for transparent and easily reproducible comparisons of VaR constrained portfolio selection problems in Python.

In this paper, we revisit the classical problem of solving over-determined systems of nonsmooth equations numerically. We suggest a nonsmooth Levenberg--Marquardt method for its solution which, in contrast to the existing literature, does not require local Lipschitzness of the data functions. This is possible when using Newton-differentiability instead of semismoothness as the underlying tool of generalized differentiation. Conditions for fast local convergence of the method are given. Afterwards, in the context of over-determined mixed nonlinear complementarity systems, our findings are applied, and globalized solution methods, based on a residual induced by the maximum and the Fischer--Burmeister function, respectively, are constructed. The assumptions for fast local convergence are worked out and compared. Finally, these methods are applied for the numerical solution of bilevel optimization problems. We recall the derivation of a stationarity condition taking the shape of an over-determined mixed nonlinear complementarity system involving a penalty parameter, formulate assumptions for local fast convergence of our solution methods explicitly, and present results of numerical experiments. Particularly, we investigate whether the treatment of the appearing penalty parameter as an additional variable is beneficial or not., Comment: 41 pages, 4 figures, 2 tables

Surges that have been observed at different periods in the number of COVID-19 cases are associated with the emergence of multiple SARS-CoV-2 (Severe Acute Respiratory Virus) variants. The design of methods to support laboratory detection are crucial in the monitoring of these variants. Hence, in this paper, we develop a semi-automated method to design both forward and reverse primer sets to detect SARS-CoV-2 variants. To proceed, we train deep Convolution Neural Networks (CNNs) to classify labelled SARS-CoV-2 variants and identify partial genomic features needed for the forward and reverse Polymerase Chain Reaction (PCR) primer design. Our proposed approach supplements existing ones while promoting the emerging concept of neural network assisted primer design for PCR. Our CNN model was trained using a database of SARS-CoV-2 full-length genomes from GISAID and tested on a separate dataset from NCBI, with 98\% accuracy for the classification of variants. This result is based on the development of three different methods of feature extraction, and the selected primer sequences for each SARS-CoV-2 variant detection (except Omicron) were present in more than 95 \% of sequences in an independent set of 5000 same variant sequences, and below 5 \% in other independent datasets with 5000 sequences of each variant. In total, we obtain 22 forward and reverse primer pairs with flexible length sizes (18-25 base pairs) with an expected amplicon length ranging between 42 and 3322 nucleotides. Besides the feature appearance, in-silico primer checks confirmed that the identified primer pairs are suitable for accurate SARS-CoV-2 variant detection by means of PCR tests., Comment: 6 figures and 2 tables

Subcutaneous Implantable Cardioverter-Defibrillators (S-ICDs) are used for prevention of sudden cardiac death triggered by ventricular arrhythmias. T Wave Over Sensing (TWOS) is an inherent risk with S-ICDs which can lead to inappropriate shocks. A major predictor of TWOS is a high T:R ratio (the ratio between the amplitudes of the T and R waves). Currently patients' Electrocardiograms (ECGs) are screened over 10 seconds to measure the T:R ratio, determining the patients' eligibility for S-ICD implantation. Due to temporal variations in the T:R ratio, 10 seconds is not long enough to reliably determine the normal values of a patient's T:R ratio. In this paper, we develop a convolutional neural network (CNN) based model utilising phase space reconstruction matrices to predict T:R ratios from 10-second ECG segments without explicitly locating the R or T waves, thus avoiding the issue of TWOS. This tool can be used to automatically screen patients over a much longer period and provide an in-depth description of the behaviour of the T:R ratio over that period. The tool can also enable much more reliable and descriptive screenings to better assess patients' eligibility for S-ICD implantation.

We consider the optimistic bilevel optimization problem, known to have a wide range of applications in engineering, that we transform into a single-level optimization problem by means of the lower-level optimal value function reformulation. Subsequently, based on the partial calmness concept, we build an equation system, which is parameterized by the corresponding partial exact penalization parameter. We then design and analyze a Levenberg-Marquardt method to solve this parametric system of equations. Considering the fact that the selection of the partial exact penalization parameter is a critical issue when numerically solving a bilevel optimization problem, we conduct a careful experimental study to this effect, in the context the Levenberg-Marquardt method, while using the Bilevel Optimization LIBrary (BOLIB) series of test problems., Comment: 21 pages, 14 figures

We consider a bilevel program involving a linear lower level problem with left-hand-side perturbation. We then consider the Karush-Kuhn-Tucker reformulation of the problem and subsequently build a tractable optimization problem with linear constraints by means of a partial exact penalization. A semismooth system of equations is then generated from the later problem and a Newton-type method is developed to solve it. Finally, we illustrate the convergence and practical implementation of the algorithm on the optimal toll-setting problem in transportation networks.

This paper considers repeated games in which one player has more information about the game than the other players. In particular, we investigate repeated two-player zero-sum games where only the column player knows the payoff matrix A of the game. Suppose that while repeatedly playing this game, the row player chooses her strategy at each round by using a no-regret algorithm to minimize her (pseudo) regret. We develop a no-instant-regret algorithm for the column player to exhibit last round convergence to a minimax equilibrium. We show that our algorithm is efficient against a large set of popular no-regret algorithms of the row player, including the multiplicative weight update algorithm, the online mirror descent method/follow-the-regularized-leader, the linear multiplicative weight update algorithm, and the optimistic multiplicative weight update., Comment: Accepted at International Conference on Algorithmic Learning Theory, PMLR 132:553-577, 2021

Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush--Kuhn--Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying issues with the aid of illustrative counterexamples., Comment: 16 pages, 1 figure

We address the problem of predicting the occurrence of infrequent adverse events in the context of predictive maintenance. We cast the corresponding machine learning task as an imbalanced classification problem and propose a framework for solving it that is capable of leveraging different classifiers in order to predict the occurrence of an adverse event before it takes place. In particular, we focus on two applications arising in low-carbon energy production: foam formation in anaerobic digestion and condenser tube leakage in the steam turbines of a nuclear power station. The results of an extensive set of omputational experiments show the effectiveness of the techniques that we propose.

We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization problem with a penalization of the value function constraint. For treating the latter problem, we develop a framework that does not rely on the direct computation of the lower-level value function or its derivatives. For each penalty parameter, the framework leads to a semismooth system of equations. This allows us to extend the semismooth Newton method to bilevel optimization. Besides global convergence properties of the method, we focus on achieving local superlinear convergence to a solution of the semismooth system. To this end, we formulate an appropriate CD-regularity assumption and derive suffcient conditions so that it is fulfilled. Moreover, we develop conditions to guarantee that a solution of the semismooth system is a local solution of the bilevel optimization problem. Extensive numerical experiments on $124$ examples of nonlinear bilevel optimization problems from the literature show that this approach exhibits a remarkable performance, where only a few penalty parameters need to be considered., Comment: 27 pages, 2 tables, 1 figure

This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial problem data. This is done by exploiting several different reformulations of the hierarchical model as a single-level problem. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel optimization problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial problem data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions., Comment: 30 pages

This chapter presents the Bilevel Optimization LIBrary of the test problems (BOLIB for short), which contains a collection of test problems, with continuous variables, to help support the development of numerical solvers for bilevel optimization. The library contains 173 examples with 138 nonlinear, 24 linear, and 11 simple bilevel optimization problems. This BOLIB collection is probably the largest bilevel optimization library of test problems. Moreover, as the library is computationenabled with the MATLAB m-files of all the examples, it provides a uniform basis for testing and comparing algorithms. The library, together with all the related codes, is freely available at biopt.github.io/bolib., Comment: Second version of the library; 17 pages, 1 figure

We study a robust toll pricing problem where toll setters and users have different level of information when taking their decisions. Toll setters do not have full information on the costs of the network and rely on historical information when determining toll rates, whereas users decide on the path to use from origin to destination knowing toll rates and having, in addition, more accurate traffic data. Toll setters often also face constraints on price experimentation which means less opportunity for price revision. Motivated by this we propose a novel robust pricing methodology for fixing prices where we take non-adversarial view of nature different from the existing robust approaches. We show that our non-adversarial robustness results in less conservative pricing decisions compared to traditional adversarial nature setting. We start by first considering a single origin-destination parallel network in this new robust setting and formulate the robust toll pricing problem as a distributionally robust optimization problem, for which we develop an exact algorithm based on a mixed-integer programming formulation and a heuristic based on two-point support distribution. We further extend our formulations to more general networks and show how our algorithms can be adapted for the general networks. Finally, we illustrate the usefulness of our approach by means of numerical experiments both on randomly generated networks and on the data recorded on the road network of the city of Chicago., Comment: 28 pages

The authors' paper in Optimization 63 (2014), 505-533, see Ref. [5], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analyzed in SIAM J. Optim. 22 (2012), 1309-1343; see Ref. [4], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve functions that are nondifferentiable at some points of their domain, the main goal of the current paper is extending the two-level value function approach to deriving new necessary optimality conditions for both optimistic and pessimistic versions in bilevel programming with nonsmooth data., Comment: 18 pages

The optimal value function is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the cost/revenue generated while minimizing/maximizing a given function under some constraints. In the context of stability/sensitivity analysis, a large number of publications have been dedicated to the study of continuity and differentiability properties of the optimal value function. The differentiability aspect of works in the current literature has mostly been limited to first order analysis, with focus on estimates of its directional derivatives and subdifferentials, given that the function is typically nonsmooth. With the progress made in the last two to three decades in major subfields of optimization such as robust, minmax, semi-infinite and bilevel optimization, and their connection to the optimal value function, there is a crucial need for a second order analysis of the generalized differentiability properties of this function. This type of analysis will enable the development of robust solution algorithms, such as the Newton method. The main goal of this paper is to provide results in this direction. In fact, we derive estimates of the generalized Hessian for the optimal value function. Our results are based on two handy tools from parametric optimization, namely the optimal solution and Lagrange multiplier mappings, for which completely detailed estimates of their generalized derivatives are either well-known or can easily be obtained., Comment: 26 pages