The Submodular Bin Packing (SMBP) problem asks for packing unsplittable items into a minimal number of bins for which the capacity utilization function is submodular. SMBP is equivalent to chance-constrained and robust bin packing problems under various conditions. SMBP is a hard binary nonlinear programming optimization problem. In this paper, we propose a branch-and-price algorithm to solve this problem. The resulting price subproblems are submodular knapsack problems, and we propose a tailored exact branch-and-cut algorithm based on a piece-wise linear relaxation to solve them. To speed up column generation, we develop a hybrid pricing strategy to replace the exact pricing algorithm with a fast pricing heuristic. We test our algorithms on instances generated as suggested in the literature. The computational results show the efficiency of our branch-and-price algorithm and the proposed pricing techniques.

EUROPT, the Continuous Optimization working group of EURO, celebrated its 20 years of activity in 2020. We trace the history of this working group by presenting the major milestones that have led to its current structure and organization and its major trademarks, such as the annual EUROPT workshop and the EUROPT Fellow recognition. • We present a brief overview of the milestones of EUROPT's history, and its rich scientific activities. • We trace the main milestones which have led to the current structure and organization of EUROPT as well as to define its main trademarks. • We trace the history of the annual EUROPT workshop and of the EUROPT Fellow award.

• We survey mixed-integer programming techniques as they are applied in bilevel optimization. • We focus on bilevel problems with convex or linear lower-level problems as well as on problems with mixed-integer lower levels. • Special attention is given to bilevel pricing problems, Stackelberg games, and interdiction problems. • The survey contains more than 250 references. Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980’s but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.

• we introduce the Steiner Bi-objective Shortest Path Problem. • We propose a solution method based on a labeling approach with a multi-objective. • Test instances are generated from real road networks. • Computational results show the efficiency of the proposed algorithm compared to state-of-art approaches. In this paper, we introduce the Steiner Bi-objective Shortest Path Problem. This problem is defined on a directed graph G = ( V , A ) , with a subset T ⊂ V of terminals. Arcs are labeled with travel time and cost. The goal is to find a complete set of efficient paths between every pair of nodes in T . The motivation behind this problem stems from data preprocessing for vehicle routing problems. We propose a solution method based on a labeling approach with a multi-objective A* search strategy guiding the search towards the terminals. Computational results based on instances generated from real road networks show the efficiency of the proposed algorithm compared to state-of-art approaches.

• A globally convergent evolution strategy to solve general constrained optimization problems. • Quantifiable relaxable constraints are handled using a merit function approach combined with a specific restoration procedure. • The proposed approach is very competitive compared to existing derivative-free optimization solvers on a large set of problems. In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles quantifiable relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints, when present, can be treated either by using the extreme barrier function or through a projection approach. Under reasonable assumptions, the introduced extension guarantees to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Numerical experiments are carried out on a set of problems from the CUTEst collection as well as on known global optimization problems.

The location of shelters in different areas threatened by wildfires is one of the possible ways to reduce fatalities in a context of an increasing number of catastrophic and severe wildfires. These shelters will enable the population in the area to be protected in case of fire outbreaks. The subject of our study is to determine the best place for shelters in a given territory. The territory, divided into zones, is represented by a graph in which each zone corresponds to a node and two nodes are linked by an edge if it is feasible to go directly from one zone to the other. The problem is to locate p shelters on nodes so that the maximum distance of any node to its nearest shelter is minimized. When the uncertainty of fire outbreaks is not considered, this problem corresponds to the well-known p-Center problem on a graph. In this article, the uncertainty of fire outbreaks is introduced taking into account a finite set of fire scenarios. A scenario defines a fire outbreak on a single zone with the main consequence of modifying evacuation paths. Several evacuation paths may become impracticable and the ensuing evacuation decisions made under pressure may no longer be rational. In this context, the new issue under consideration is to place p shelters on a graph so that the maximum evacuation distance of any node to its nearest shelter in any scenario is minimized. We refer to this problem as the Robust p-Center problem under Pressure. After proving the NP-hardness of this problem on subgraphs of grids, we propose a first formulation based on 0-1 Linear Programming. For real size instances, the sizes of the 0-1 Linear Programs are huge and we propose a decomposition scheme to solve them exactly. Experimental results outline the efficiency of our approach.

Multistage stochastic programs arise in many applications from engineering whenever a set of inventories or stocks has to be valued. Such is the case in seasonal storage valuation of a set of cascaded reservoir chains in hydro management. A popular method is stochastic dual dynamic programming (SDDP), especially when the dimensionality of the problem is large and dynamic programming is no longer an option. The usual assumption of SDDP is that uncertainty is stage-wise independent, which is highly restrictive from a practical viewpoint. When possible, the usual remedy is to increase the state-space to account for some degree of dependency. In applications, this may not be possible or it may increase the state-space by too much. In this paper, we present an alternative based on keeping a functional dependency in the SDDP—cuts related to the conditional expectations in the dynamic programming equations. Our method is based on popular methodology in mathematical finance, where it has progressively replaced scenario trees due to superior numerical performance. We demonstrate the interest of combining this way of handling dependency in uncertainty and SDDP on a set of numerical examples. Our method is readily available in the open-source software package StOpt.

The plain Newton-min algorithm for solving the linear complementarity problem (LCP) “ 0 ⩽ x ⊥ ( M x + q ) ⩾ 0 ” can be viewed as an instance of the plain semismooth Newton method on the equational version “ min ( x , M x + q ) = 0 ” of the problem. This algorithm converges for any q when M is an M -matrix, but not when it is a P -matrix. When convergence occurs, it is often very fast (in at most n iterations for an M -matrix, where n is the number of variables, but often much faster in practice). In 1990, Harker and Pang proposed to improve the convergence ability of this algorithm by introducing a stepsize along the Newton-min direction that results in a jump over at least one of the encountered kinks of the min-function, in order to avoid its points of nondifferentiability. This paper shows that, for the Fathi problem (an LCP with a positive definite symmetric matrix M , hence a P -matrix), an algorithmic scheme, including the algorithm of Harker and Pang, may require n iterations to converge, depending on the starting point.

The resource-constrained project scheduling problem (RCPSP), one of the most challenging combinatorial optimization scheduling problems, has been the focus of a great deal of research, resulting in numerous publications in the last decade. Previous publications focused on the RCPSP, including several extensions with different objectives to be minimized and constraints to be checked. The present work investigates the integration of the routing, i.e., the transport of the resources between activities into the RCPSP, and provides a new resolution scheme. The two subproblems are solved using an integrated approach that draws on both a disjunctive graph model and an explicit modeling of the routing. The resolution scheme takes advantage of an indirect representation of the solution to define both the schedule of activities and the routing of vehicles. The routing solution is modeled by a set of trips that define the loaded transport operations of vehicles that are induced by the flow in the graph. The numerical experiments prove that the models and the methods introduced in this paper are promising for solving the RCPSP with routing.

In the two-dimensional guillotine cutting-stock problem, the objective is to minimize the number of large plates used to cut a list of small rectangles. We consider a variant of this problem, which arises in glass industry when different bills of order (or batches) are considered consecutively. For practical organization reasons, leftovers are not reused, except the large one obtained in the last cutting pattern of a batch, which can be reused for the next batch. The problem can be decomposed into an independent problem for each batch. In this paper, we focus on the one-batch problem, the objective of which is to minimize the total width of the cutting patterns used. We propose a diving heuristic based on column generation, in which the pricing problem is solved using dynamic programming (DP). This DP generates so-called non-proper columns, i.e. cutting patterns that cannot participate in a feasible integer solution of the problem. We show how to adapt the standard diving heuristic to this “non-proper” case while keeping its effectiveness. We also introduce the partial enumeration technique, which is designed to reduce the number of non-proper patterns in the solution space of the dynamic program. This technique strengthens the lower bounds obtained by column generation and improves the quality of the solutions found by the diving heuristic. Computational results are reported and compared on classical benchmarks from the literature as well as on new instances inspired from glass industry data. According to these results, variants of the proposed diving heuristic outperform constructive and evolutionary heuristics.

The minimum labeling spanning tree problem (MLSTP) is a combinatorial optimization problem that consists in finding a spanning tree in a simple graph G, in which each edge has one label, by using a minimum number of labels. It is an NP-hard problem that was introduced by Chang and Leu (Inf Process Lett 63(5):277–282, 1997. https://doi.org/10.1016/S0020-0190(97)00127-0 ). Chen et al. (Comparison of heuristics for solving the gmlst problem, in: Raghavan, Golden, Wasil (eds) Telecommunications modeling, policy, and technology, Springer, Boston, pp 191–217, 2008) subsequently proposed a generalization of the MLSTP, called the generalized minimum labeling spanning tree problem (GMLSTP), that allows a situation in which multiple labels can be assigned to an edge. Here, we show how the GMLSTP can be expressed as an MLSTP in a multigraph. Both problems have applications in various areas such as computer network design, multimodal transportation network design, and data compression. We propose a new compact binary integer programming model to solve exactly the GMLSTP and analyze the polytope associated with the formulation. The paper introduces new concepts, gives the polytope dimension, and describes five new facet families. The polyhedral comparison results for the studied polytope show that the new model is theoretically superior to current state-of-the-art formulations.

We consider linear programs involving uncertain parameters and propose a new tractable robust counterpart which contains and generalizes several other models including the existing Affinely Adjustable Robust Counterpart and the Fully Adjustable Robust Counterpart . It consists in selecting a set of poles whose convex hull contains some projection of the uncertainty set, and computing a recourse strategy for each data scenario as a convex combination of some optimized recourses (one for each pole). We show that the proposed multipolar robust counterpart is tractable and its complexity is controllable. Further, we show that under some mild assumptions, two sequences of upper and lower bounds converge to the optimal value of the fully adjustable robust counterpart. We numerically investigate a couple of applications in the literature demonstrating that the approach can effectively improve the affinely adjustable policy.

This paper investigates a portfolio optimization problem under uncertainty on the stock returns, where the manager seeks to achieve an appropriate trade-off between the expected portfolio return and the risk of loss. The uncertainty set consists of a finite set of scenarios occurring with equal probability. We introduce a new robustness criterion, called pw -robustness, which seeks to maximize the portfolio return in a proportion p of scenarios and guarantees a minimum return over all scenarios. We model this optimization problem as a mixed-integer programming problem. Through extensive numerical experiments, we identify the instances that can be solved to optimality in an acceptable time using off-the-shelf software. For the instances that cannot be solved to optimality within the time frame, we propose and test a heuristic that exhibits excellent practical performance in terms of computation time and solution quality for the problems we consider. This new criterion and our heuristic methods therefore exhibit great promise to tackle robustness problems when the uncertainty set consists of a large number of scenarios.

Many practical combinatorial optimization problems can be described by integer linear programs having an exponential number of variables, and they are efficiently solved by column generation algorithms. For these problems, column generation is used to compute good dual bounds that can be incorporated in branch-and-price algorithms. Recent research has concentrated on describing lower and upper bounds of bi-objective and general multi-objective problems with sets of points (bound sets). An important issue to address when computing a bound set by column generation is how to efficiently search for columns corresponding to each point of the bound set. In this work, we propose a generalized column generation scheme to compute bound sets for bi-objective combinatorial optimization problems. We present specific implementations of the generalized scheme for the case where one objective is a min–max function by using a variant of the $$\varepsilon $$ -constraint method to efficiently model these problems. The proposed strategies are applied to a bi-objective extension of the multi-vehicle covering tour problem, and their relative performances based on different criteria are compared. The results show that good bound sets can be obtained in reasonable times if columns are efficiently managed. The variant of the $$\varepsilon $$ -constraint presented is also better than a standard $$\varepsilon $$ -constraint method in terms of the quality of the bound sets.

One challenge for social network researchers is to evaluate balance in a social network. The degree of balance in a social group can be used as a tool to study whether and how this group evolves to a possible balanced state. The solution of clustering problems defined on signed graphs can be used as a criterion to measure the degree of balance in social networks and this measure can be obtained with the optimal solution of the correlation clustering problem, as well as a variation of it, the relaxed correlation clustering problem. However, solving these problems is no easy task, especially when large network instances need to be analyzed. In this work, we contribute to the efficient solution of both problems by developing sequential and parallel ILS metaheuristics. Then, by using our algorithms, we solve the problem of measuring the structural balance on large real-world social networks.

In this paper, we address a rich Traveling Salesman Problem with Profits encountered in several real-life cases. We propose a unified solution approach based on variable neighborhood search. Our approach combines several removal and insertion routing neighborhoods and efficient constraint checking procedures. The loading problem related to the use of a multi-compartment vehicle is addressed carefully. Two loading neighborhoods based on the solution of mathematical programs are proposed to intensify the search. They interact with the routing neighborhoods as it is commonly done in matheuristics. The performance of the proposed matheuristic is assessed on various instances proposed for the Orienteering Problem and the Orienteering Problem with Time Window including up to 288 customers. The computational results show that the proposed matheuristic is very competitive compared with the state-of-the-art methods. To better evaluate its performance, we generate a new testbed including instances with various attributes. Extensive computational experiments on the new testbed confirm the efficiency of the matheuristic. A sensitivity analysis highlights which components of the matheuristic contribute most to the solution quality.

Variable neighborhood search (VNS) is a framework for building heuristics, based upon systematic changes of neighborhoods both in a descent phase, to find a local minimum, and in a perturbation phase to escape from the corresponding valley. In this paper, we present some of VNS basic schemes as well as several VNS variants deduced from these basic schemes. In addition, the paper includes parallel implementations and hybrids with other metaheuristics.

We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine's positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) In contrast to the standard SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user. (b) In contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems. Finally (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging.

This paper elaborates compact MIP formulations for a discrete unit commitment problem with minimum stop and ramping constraints. The variables can be defined in two different ways. Both MIP formulations are tightened with clique cuts and local constraints. The projection of constraints from one variable structure to the other allows to compare and tighten the MIP formulations. This leads to several equivalent formulations in terms of polyhedral descriptions and thus in LP relaxations. We analyse how MIP resolutions differ in the efficiency of the cuts, branching and primal heuristics. The resulting MIP implementation allows to tackle real size instances for an industrial application.

We consider convex nonsmooth optimization problems where additional information with uncontrolled accuracy is readily available. It is often the case when the objective function is itself the output of an optimization solver, as for large-scale energy optimization problems tackled by decomposition. In this paper, we study how to incorporate the uncontrolled linearizations into (proximal and level) bundle algorithms in view of generating better iterates and possibly accelerating the methods. We provide the convergence analysis of the algorithms using uncontrolled linearizations, and we present numerical illustrations showing they indeed speed up resolution of two stochastic optimization problems coming from energy optimization (two-stage linear problems and chance-constrained problems in reservoir management).

In the aftermath of disasters such as major earthquakes, several roads may be blocked by rubble and the population tends to search refugee in certain gathering points of the city. Road network accessibility becomes an important issue for logistic operations, specially on the first days after the quake, when the relief distribution is crucial for survival. This study focused on the Road Emergency Rehabilitation Problem, divided into the Road Network Accessibility Problem (RNAP) and the Work-troops Scheduling Problem (WSP). The first one consists in finding traversable paths for relief teams to reach the population, and the later generates a repairing schedule to improve access to refugee areas. The contributions of this study are two-fold: we present the process of transcribing satellite imagery data into graphs, and mathematical formulations for the RNAP and WSP, along with heuristics to solve the WSP. The proposed methods are able to handle large-scale graphs in an acceptable running time for real scenarios. They are tested on simulated instances and on the graph of Port-au-Prince, with more than 10,000 vertices and edges. The Port-au-Prince graph was generated from satellite images obtained by the International Charter “Space and Major Disasters” a few hours after the 2010 earthquake.

We consider a network flow problem, where the outgoing flow is reduced by a certain percentage in each node. Given a maximum amount of flow that can leave the source node, the aim is to find a solution that maximizes the amount of flow which arrives at the sink. Starting from this basic model, we include two new, additional aspects: On the one hand, we are able to reduce the loss at some of the nodes; on the other hand, the exact loss values are not known, but may come from a discrete uncertainty set of exponential size. Applications for problems of this type can be found in evacuation planning, where one would like to improve the safety of nodes such that the number of evacuees reaching safety is maximized. We formulate the resulting robust flow problem with losses and improvability as a two-stage mixed-integer program with uncertain recourse for finitely many scenarios and present an iterative scenario-generation procedure that avoids the inclusion of all scenarios from the beginning as well as several heuristic solution methods. In a computational study using both randomly generated instances and realistic data based on the city of Nice, France, we compare our solution algorithms.