In this paper we define the first class of algebraically equivalent transformation (AET) functions for predictor-corrector (PC) interior-point algorithms (IPAs) for sufficient linear complementarity problems (LCPs) and we provide the unified complexity analysis of the introduced PC IPA. This new class is called the class of AET functions with bounded proportional growth rate for PC IPAs. The main difference between our PC IPA and those from the literature is that the AET function is an input data of our algorithm. The new class of AET functions is defined by several inequalities that are directly used in the analysis of the new PC IPA. We show that the PC IPA using any function of the new class of AET functions has polynomial iteration complexity in the size of the problem, the starting point’s duality gap, the accuracy parameter and the handicap of the problem’s matrix. We also provide promising numerical results on a newly generated test set of problems that contain matrices with large handicap, as well.
Hierarchical Nash game models such as Multi-Leader-Follower games have recently attracted the interest of research as they are an important modeling tool in various applications to study a strategic decision process of individuals, where the individuals are split into a hierarchy of different groups. In this paper, we address these game models using a parametric optimization approach in order to reduce the number of levels to a single-level game. In particular, we further consider the potential followers game to reduce the complexity of the problem. On the one hand, we study various mathematical structures and properties of such nonlinear hierarchical games. On the other hand, we provide suitable numerical solution approaches for the resulting single-level, but nonconvex, non-smooth or disjunctive optimization problems based on the specific theoretical properties. Finally, we finish the paper with computational results for some numerical experiments that support our approaches.
The security-constrained unit commitment problem seeks the minimum cost schedule of electricity generation units that meets demand over a 24h horizon and remains robust to failures of the transmission branches of the network linking generators and loads. Due to the size of realistic problems, which involve millions of variables and constraints, state-of-the-art solution techniques rely on linearizing the physical laws governing energy flows through branches and applying decomposition methods. We formulate this problem using high fidelity second order conic expressions for energy flows, yielding a large-scale mixed integer second order conic problem. We solve it using a decomposition procedure that separates the problem into a master problem and a large number of subproblems, and that combines Benders and column and constraint generation ideas, along with an accurate outer linearization for the master problem. We conduct computational experiments on realistic case studies to demonstrate the efficiency of the proposed methodology. For a number of realistic instances, we compare our results with those obtained using state-of-the-art commercial solvers.
Counterfactual explanations (CEs) are powerful tools within the thriving field of explainable machine learning, consisting of finding a minimal perturbation to a given instance so that its output in a certain predictive model satisfies some desirable property. Since counterfactuals are computed from an empirical model trained on a dataset, it is not evident that the resulting counterfactual explanations satisfy the requirements of the ground-truth model, which may differ from the empirical one. In this paper, we address this issue and build robust counterfactuals, in the sense that they fulfill the desired property on the ground-truth model with a given probability. We propose a statistical test to obtain performance guarantees for our robust counterfactual explanations, not only under the empirical model, but also under the ground-truth model. We particularize to generalized linear models (GLMs), a broad family of models that includes both classification, such as the logistic regression model, and regression, such as the linear regression model. We show that, for GLMs, there is a strong connection between CEs with ground-truth model guarantees and parameter-robust CEs. Finding a robust counterfactual explanation in the GLM framework amounts to solving a second-order cone program, readily solvable by off-the-shelf solvers such as Gurobi.
The resource-constrained project scheduling problem aims to schedule project activities given precedence relations and limited resources. This well-known scheduling problem has been extended into the so-called resource-constrained project scheduling problem with alternative subgraphs (RCPSP-AS). In this variant, the project network comprises multiple alternatives for executing work packages, which are smaller parts of the project, with the challenge of selecting and scheduling some of these alternatives such that the project makespan is minimised. This study introduces cost objectives and cumulative resource constraints in the RCPSP-AS, leading to the formulation of a new problem referred to as the RCPSP-ASC, where C signifies the incorporation of new cost objectives. To address this problem, novel priority rules are devised, and existing ones are adjusted. We present three empirical production case studies to underscore the relevance of these problem extensions in real-life settings. Moreover, a proposed mixed-integer mathematical model is validated using Gurobi and the computational efficiency of obtaining (near-)optimal solutions is investigated. Using existing project complexity indicators, we show that time- and resource-based priority rules are competitive for projects with low complexity. Moreover, the time-based priority rule is the best-performing rule for time minimisation and an additional combined heuristic is proposed for further improving the solutions. We also test the cost performance of a dynamic genetic algorithm (GA) from the existing literature on a subset of project instances for which the GA produces either low-quality or high-quality results in terms of time performance. Finally, an off-the-shelf constraint programming (CP) satisfiability (SAT) solver is implemented that shows improved results for the high-quality subset of instances compared to the dynamic GA and comparable results for the low-quality subset.
In recent years, the monotone inclusion problem has attracted significant research attention due to its wide range of applications. However, many existing results in the literature require the single-valued operator in the problem formulation to be co-coercive (inverse strongly monotone) and are often restricted to Hilbert spaces. This limitation reduces the applicability of such results, as real-world problems involving monotone operators are not uncommon. In this paper, we study a class of monotone inclusion problems where the single-valued operator is not necessarily co-coercive, together with the common fixed point problem for relatively nonexpansive multivalued mappings. We propose a new iterative method for solving this common solution problem in the framework of Banach spaces. Our method incorporates several techniques to achieve high computational efficiency. One key component is the S-iteration process, which is known to outperform many classical methods. In addition, the proposed algorithm employs an inertial technique and a non-monotonic self-adaptive step size to ensure a high rate of convergence and ease of implementation. We establish strong convergence results for the proposed method under mild conditions and apply our findings to study certain optimization problems. Finally, we demonstrate the practical efficiency of the proposed method through extensive numerical experiments on real-world applications, including image restoration, economic modelling via price adjustment under uncertainty, and robust consensus in networked control systems. Our results extend and improve upon several recently published works in this area.
A widely used approximation concept in multiobjective optimization is the concept of enclosures. These are unions of boxes defined by lower and upper bound sets that are used to cover optimal sets of multiobjective optimization problems in the image space. The width of an enclosure is taken as a quality measure. In this paper, we provide properties of enclosures and their width in multiobjective optimization. To apply enclosures for warm-start strategies and for approximations of optimal sets, we discuss under which conditions enclosures have nonempty interior and whether they coincide with the closure of their interior. We extend the optimality concepts of E-minimality and E-weak minimality from multiobjective optimization and introduce new optimality concepts for multiobjective optimization caused by relaxations of multiobjective optimization problems. We show that the enclosures and their widths are suitable for determining these new optimal points. We provide some calculation and estimation rules for the width of enclosures, such as a monotonicity, decomposition, and combination property and a triangular inequality-like relation. These are important for convergence examinations for approximation algorithms. All our examinations are driven by the requirements to further explore enclosure-based algorithms for the numerical computation of optimal points in multiobjective optimization. Hence, this paper provides a toolbox of theoretical results for enclosures that supports the development of convergence proofs of image-space-based approximation methods for several classes of multiobjective optimization problems.
This paper revisits and extends the 2013 development by Rockafellar and Uryasev of the Risk Quadrangle (RQ) as a unified scheme for integrating risk management, optimization, and statistical estimation. The RQ features four stochastic-oriented functionals-risk, deviation, regret, and error, along with an associated statistic, and articulates their revealing and in some ways surprising interrelationships and dualizations. Additions to the RQ framework that have come to light since 2013 are reviewed in a synthesis focused on both theoretical advancements and practical applications. New quadrangles-superquantile, superquantile norm, expectile, biased mean, quantile symmetric average union, and p-divergence-based quadrangles-offer novel approaches to risk-sensitive decision-making across various fields such as machine learning, statistics, finance, and PDE-constrained optimization. The theoretical contribution comes in axioms for "subregularity" relaxing "regularity" of the quadrangle functionals, which is too restrictive for some applications. The main RQ theorems and connections are revisited and rigorously extended to this more ample framework. Examples are provided in portfolio optimization, regression, and classification, demonstrating the advantages and the role played by duality, especially in ties to robust optimization and generalized stochastic divergences.
This paper investigates the integration of machine learning forecasts of intervention durations into a stochastic variant of the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW). In particular, we exploit tree-based gradient boosting (XGBoost) trained on eight years of gas meter maintenance data to produce point predictions and uncertainty estimates, which then drive a multi-objective evolutionary optimization routine. The methodology addresses uncertainty through sub-Gaussian concentration bounds for route-level risk buffers and explicitly accounts for competing operational KPIs through a multi-objective formulation. Empirical analysis of prediction residuals validates the sub-Gaussian assumption underlying the risk model. From an empirical point of view, our results report improvements of around 20-25% in operator utilization and completion rates compared with plans computed using default durations and unconditional stochastic baselines. The integration of uncertainty quantification and risk-aware optimization provides a practical framework for handling stochastic service durations in real-world routing applications.
The concept of local upper bounds plays an important role in numerical algorithms for nonconvex, integer, and mixed-integer multiobjective optimization with respect to the componentwise partial ordering, that is, where the ordering cone is the nonnegative orthant. In this paper, we answer the question of whether and how this concept can be extended to arbitrary ordering cones. We define local upper bounds with respect to a closed pointed solid convex cone and study their properties. We show that for special polyhedral ordering cones the concept of local upper bounds can be as practical as it is for the nonnegative orthant.
The Maximum Diversity Problem (MDP) is a challenging NP-hard optimization problem with applications in various domains, including social network analysis, bioinformatics, and facility location. Traditional heuristics often struggle to find high-quality solutions for large-scale MDP instances within reasonable time limits. Recent hybrid heuristics have incorporated data mining techniques to guide the search process, yielding improved solution quality. MineReduce is a successful example of such recent proposals. Its approach leverages mined patterns to contract portions of the problem space, facilitating a more focused and efficient search. In this work, we propose a new heuristic that integrates the MineReduce technique with the MDM_KLD, a previously proposed hybrid heuristic, to solve the MDP. Our approach aims to alleviate the computational burden by periodically solving a reduced version of the problem without compromising the quality of the solutions obtained for the original problem. Computational experiments conducted on three different sets of instances demonstrate the effectiveness of our approach compared to MDM_KLD, achieving superior solution quality in most instances within the same computational time.
We address the challenge of efficiently solving parameterized sequences of convex Mixed-Integer Nonlinear Programming (MINLP) problems through warm-starting techniques. We focus on an outer approximation (OA) approach, for which we develop the theoretical foundation and present two warm-starting techniques for solving sequences of convex MINLPs. These types of problem sequences arise in several important applications, such as, multiobjective MINLPs using scalarization techniques, sparse linear regression, hybrid model predictive control, or simply in analyzing the impact of certain problem parameters. The main contribution of this paper is the mathematical analysis of the proposed warm-starting framework for OA-based algorithms, which shows that a simple adaptation of the linear relaxation from one problem to the next can greatly improve computational performance. In the case that the parameters depend linearly on the parameter, we prove under some assumptions that one of the proposed warm-starting techniques results in only one OA iteration to find an optimal solution and verify optimality. Numerical results demonstrate noticeable performance improvements compared to two common initialization approaches, and show that the warm-starting can also in practice result in a single iteration to converge for several problems in the sequences. Our methods are especially effective for problems where consecutive problems in the sequence are similar, and where the integer part of the optimal solutions remains constant for several problems in the sequence. The results show that it is possible, both in theory and practice, to perform warm-starting to significantly enhance the computational efficiency of solving parameterized convex MINLPs.
A trajectory-following primal–dual interior-point method solves nonlinear optimization problems with inequality and equality constraints by approximately finding points satisfying perturbed Karush–Kuhn–Tucker optimality conditions for a decreasing order of perturbation controlled by the barrier parameter. Under some conditions, there is a unique local correspondence between small residuals of the optimality conditions and points yielding that residual, and the solution on the barrier trajectory for the next barrier parameter can be approximated using an approximate solution for the current parameter. A framework using higher-order derivative information of the correspondence is analyzed in which an extrapolation step to the trajectory is first taken after each decrease of the barrier parameter upon reaching a sufficient approximation. It suffices asymptotically to only take extrapolation steps for convergence at the rate the barrier parameter decreases with when using derivative information of high enough order. Numerical results for nonlinear programming problems are presented using extrapolation as accelerator.
A new, fast second-order method is proposed that achieves the optimal O (| log(e)|e-3/2) complexity to obtain first-order e-stationary points. Crucially, this is deduced without assuming the standard global Lipschitz Hessian continuity condition, but only using an appropriate local smoothness requirement. The algorithm exploits Hessian information to compute a Newton step and a negative curvature step when needed, in an approach similar to that of [18]. Inexact versions of the Newton step and negative curvature are proposed in order to reduce the cost of evaluating second-order information. Details are given of such an iterative implementation using Krylov subspaces. An extended algorithm for finding second-order critical points is also developed and its complexity is again shown to be within a log factor of the optimal one. Initial numerical experiments are discussed for both factorized and Krylov variants, which demonstrate the competitiveness of the proposed algorithm.
We introduce an extension of the Difference of Convex Algorithm (DCA) in the form of a randomized block coordinate approach for problems with separable structure. For n coordinate-blocks and k iterations, our main result proves a non-asymptotic convergence rate of O(n/k) in expectation, with respect to a stationarity measure based on a Forward-Backward envelope. Furthermore, leveraging the connection between DCA and Expectation Maximization (EM), we propose a randomized block coordinate EM algorithm.
In this paper, we investigate two known solution approaches for set-valued optimization problems, both of which are based on so-called vectorization strategies. These strategies consist of deriving a parametric family of multi-objective optimization problems whose optimal solution sets approximate those of the original set-valued problem with arbitrary accuracy in a certain sense. Thus, these approaches can serve as a basis for the numerical solution of set-valued optimization problems using established solution algorithms from multi-objective optimization. We show that many properties that have already been obtained for one of the two vectorization schemes also hold for the other similarly. Thereby, it turns out that under certain assumptions there exist problem classes for both vectorization schemes in which the set-valued initial problems are even equivalent to the corresponding multi-objective replacement problems. This property is fulfilled, for example, for set-valued optimization problems with a finite feasible set, a polytope-valued objective map, or a convex graph. This was already known for one of the two vectorization schemes, and could now also be shown for the other scheme.
Recent advances in deep learning techniques pose a question of whether they can facilitate the task of finding good quality solutions to combinatorial optimization (CO) problems in a practically relevant solution time. Specifically, it is of practical relevance to determine to what extent graph neural networks (GNNs) can be applied to CO problems that can be formulated as QUBOs and thus be naturally interpreted as graph problems. In this research a GNN solver is applied to two classical CO problems-the maximum cut problem and maximum independent set problem-in an unsupervised learning setting. We show that while GNN solver consistently finds good quality solutions for the Max Cut problem irrespective of the size and density of the graph, solving MIS problems is challenging for all but very sparse graphs. We further show how this problem can be addressed by embedding transfer between these two problems and compare two different GNN architectures-GCN and GraphSAGE on their robustness with respect to graph density and symmetry. Finally we demonstrate that changing the widely used Adam optimizer to Rprop optimizer can lead to considerable reduction in solution times.
This paper addresses with minimizing the makespan in unrelated parallel machines with sequence and machine-dependent setup times involving a non-overlapping shared resource. The shared resource corresponds to an operator who can service one machine at a time to perform setup operations when it is necessary to change the type of job to be processed. We develop two Mixed Integer Linear Programming (MILP) formulations and a metaheuristic algorithm that combines an Iterated Greedy scheme with Variable Neighborhood Descent (IG + VND). Both approaches were compared with existing models and algorithms from the literature. The results indicate that the proposed formulation outperforms previous models present in the literature in instances of up to 14 jobs and 4 machines. The algorithm reported average deviations within 5 % of the solutions obtained by the model, but performed up to 3 orders of magnitude faster. Similarly, the proposed IG+VND outperformed the other algorithms in the literature, particularly for instances of up to 40 jobs and 6 machines, and obtained solutions for instances of up to 60 jobs and up to 6 machines.
Presolve techniques have been experimentally shown to significantly accelerate the performance of optimization solvers, achieving speedups of several orders of magnitude in widely used benchmarks. Building on the success of these techniques in linear and mixed-integer linear optimization problems, we introduce novel presolve methods specifically designed for nonlinear optimization. These methods aim to reduce model size and nonlinearity while preserving convexity and ensuring global optimality. We propose a combined linear and nonlinear presolve approach that integrates classical linear presolve strategies with novel methods for reformulating nonlinear expressions and simplifying models. For instance, monotonicity arguments are used to fix nonlinear variables, and linear constraints are exploited to tighten bilinear products. Computational experiments on diverse nonlinear benchmarks and continuous relaxations of discrete nonlinear problems demonstrate the efficacy of our approach. The results show that the proposed methods significantly enhance the performance of one global and four local nonlinear optimization solvers.
