## Section: New Results

### Large scale complex structure optimization

**New decomposition methods for the time-dependent combined network design and routing problem:** A significant amount of work has been focussed on the design of telecommunication networks. The performance of different Integer Programming models for various situations has been computationally assessed. One of the settings that has been thoroughly analyzed is a variant where routing decisions (for time-dependent traffic demand), and network design, are combined in a single optimization model. Solving this model with a state-of-the-art solver on representative network topologies, shows that this model quickly becomes intractable. With an extended formulation, both the number of continuous flow variables and the number of fixed charge capacity constraints are multiplied by a factor $\left|V\right|$ (where $V$ represents the set of nodes) leading to large model. However, the linear relaxation of this extended formulation yields much better lower bounds. Nevertheless, even if the extended model provides stronger lower bounds than the aggregated formulation, it suffers from its huge size: solving the linear relaxation of the problem quickly becomes intractable when the network size increases, making the linear relaxation expensive to solve. This observation motivates the analysis of decomposition methods [21].

**Convex piecewise linear unsplittable multicommodity flow problems**
We studied the multi-commodity flow problem with unsplittable flows, and
piecewise-linear costs on the arcs. They show that this problem is NP-hard when there is more
than one commodity. We propose a new MILP models for this problem, that was compared to two
formulations commonly used in the literature. The computational experiments reveal that the new model
is able to obtain very strong lower bounds, and is very efficient to solve the considered problem [22].

**Tree Reconstruction Problems: **
We studied the problem of reconstructing a tree network by knowing
only its set of terminal nodes and their pairwise distances, so that the
reconstructed network has its total edge weight minimized. This problem
has applications in several areas, namely the inference of phylogenetic
trees and the inference of routing networks topology. Phylogenetic
trees allow the understanding of the evolutionary history of species and
can assist in the development of vaccines and the study of
biodiversity. The knowledge of the routing network topology is the basis
for network tomography algorithms and it is a key strategy to the
development of more sophisticated and ambitious traffic control
protocols and dynamic routing algorithms [24].

**Distribution network configuration problems:** A distribution network is a system aiming to transfer a certain type of resource from feeders to customers. Feeders are producers of a resource and customers have a certain demand in this resource that must be satisfied. Distribution networks can be represented on graphs and be subject to constraints that limit the number of intermediate nodes between some elements of the network (hop constraints) because of physical constraints. We used layered graphs for hop constrained problems to build extended formulations [16]. Preprocessing techniques allowed to reduce the size of the layered graphs used. The model was studied on the hop-constrained minimum margin problem in an electricity network. This problem consists of designing a connected electricity distribution network, and to assign customers to electricity feeders at a maximum number of hops so as to maximize the minimum capacity margin over the feeders to avoid an overload for any feeder.

**Comparison of formulations and solution methods for location problems: **
We addressed two classes of location problems the Discrete Ordered Median Problem (DOMP) and
the two-level uncapacitated facility location problem with single assignment constraints
(TUFLP-S), an extension of the uncapacitated facility location problem.
We presented several new formulations for the DOMP based on its similarity with some
scheduling problems. Some of the new formulations present a considerably
smaller number of constraints to define the problem with respect to some previously known formulations.
Furthermore, the lower bounds provided by their linear relaxations improve the ones obtained with previous
formulations in the literature even when strengthening is not applied. We also present a polyhedral study
of the assignment polytope of our tightest formulation showing its proximity to the convex hull of the
integer solutions of the problem. Several resolution approaches, among which we mention a branch and cut
algorithm, are compared. Extensive computational results on two families of instances, namely randomly
generated and from Beasley's OR-library, show the power of our methods for solving DOMP [28].
We also addressed the TUFLP-S for which we presented six mixed-integer programming models
based on reformulation techniques and on the relaxation of the integrality of some of the variables associated with location decisions. We compared the models by carrying out extensive computational experiments on large, hard, artificial instances, as well as on instances derived from an industrial application in freight transportation [27].

**New models and algorithms for integrated vehicle routing problems: **
We address a real-life inventory routing problem, which consists in designing routes and managing the inventories of
the customers simultaneously. The problem was introduced during the 2016 ROADEF/EURO challenge.
The proposed problem is original and complex for several reasons: the logistic ratio optimization objective, the hourly time-granularity for inventory constraints over a large planning horizon, the driver/trailer allocation management. Clearly, this problem is an optimization problem with complex structure, for which we propose an extended formulation that we address with a heuristic branch-price-and-cut method.
Among the difficulties, that we had to face, are: the fractional objective function, the simultaneous generation of constraints and columns, and a complex pricing problem.
We evaluate our approach on the benchmark instances proposed by the enterprise Air Liquide co-organiser of challenge.
The solution method allowed the team including INOCS members to win the scientific prize of the ROADEF/EURO challenge 2016 [47]. We also addressed a rich Traveling Salesman Problem with Profits
encountered in several real-life cases. We proposed a unified solution approach
based on variable neighborhood search. Our approach includes 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. Extensive computational experiments on the new testbed confirm the efficiency of the matheuristic [30].

**A heuristic approach to solve an integrated warehouse order picking problem:**
We study an integrated warehouse order picking problem with manual picking operations.
The picking area of the warehouse is composed by a set of storage positions.
The working day is divided in periods.
For each period, each position contains several pieces of a unique product, and a set of customer orders has to be prepared.
An order is a set of products, each associated with a quantity.
The order pickers can prepare up to $K$ parcels in a given picking route.
The problem consists in jointly deciding:
(1) the assignment of references to storage positions in the aisles which need to be filled up;
(2) the division of orders into several parcels, respecting weight and size constraints;
(3) the batching of parcels into groups of size $K$, that implicitly define the routing into the picking area.
The objective function is to minimize the total walking distance.
In order to deal with industrial instances of large size (considering hundreds of clients, thousands of positions and product references) in a short computation time, a heuristic method based on dynamic programming and minimum cost flow paradigms is proposed.
Experimental results on an industrial benchmark have reported very good results with respect to the actual industrial solution [64].

**New models for Load Scheduling for Residential Demand Response on Smart Grids. **
The residential load scheduling problem is concerned with finding an optimal schedule for the operation of residential loads
so as to minimize the total cost of energy while aiming to respect a prescribed limit on the power level of the residence.
We propose a mixed integer linear programming formulation of this problem that accounts for the consumption of appliances,
generation from a photovoltaic system, and the availability of energy storage.
A distinctive feature of our model is the use of operational patterns that capture the individual operational characteristics of each load,
giving the model the capability to accommodate a wide range of possible operating patterns for the flexible loads.
The proposed formulation optimizes the choice of operational pattern for each load over a given planning horizon.
In this way, it generates a schedule that is optimal for a given planning horizon, unlike many alternatives based on controllers.
The formulation can be incorporated into a variety of demand response systems, in particular because it
can account for different aspects of the cost of energy, such as the cost of power capacity violations, to reflect the needs or requirements of the grid.
Our computational results show that the proposed formulation is able to
achieve electricity costs savings and to reduce peaks in the power consumption, by shifting the demand and by efficiently using a battery [52].

**Lagrangian heuristics for SVM with feature selection:**
The focus of pattern classification is to recognize similarities in the data, categorizing them in different subsets. In many fields, such as the financial and the medical ones, classification of data (samples in Machine Learning language) is useful for analysis or diagnosis purposes. Quite often datasets are formed by a small number of samples, which in turn are characterized by a huge number of attributes (features). The handling of the entire feature set would be computationally very expensive and its outcome would lack from
insight. For this reason, it is convenient to reduce the set of features which is expected to be easier to interpret and also easy to evaluate. However, it is not always easy to predict which of those are relevant for classification purposes. Hence it is necessary to screen off the relevant features from those which are irrelevant. The process that selects the features entering the subset of the relevant ones is known as Feature Selection (FS). The (FS) problem can be treated explicitly as a Mixed Binary Programming (MBP) one in the framework of the Support Vector Machine approach. We have discussed a Lagrangian-relaxation-based heuristics. In particular we embed into our objective function a weighted combination of the ${L}^{1}$ and ${L}^{0}$ norm of the normal to the separating hyperplane. We come out with a Mixed Binary Linear Programming problem which is suitable for a Lagrangian relaxation approach. Based on a property of the optimal multiplier setting, we apply a consolidated nonsmooth optimization ascent algorithm to solve the resulting Lagrangian dual. In the proposed approach we get, at every ascent step, both a lower bound on the optimal solution as well as a feasible solution at low computational cost [26].

**Decomposition methods for tree-based network design problems:** We studied different problems, where the underlying solution structure needs to have a tree-like topology and some additional constraints need to be fulfilled. For all these problems, we focused on solution approaches, which allow to tackle large-scale instances, as the application of these problems in areas like systems biology often has to deal with instances containing tens of thousands of nodes. In order to solve these problems efficiently, we turned to decomposition methods, like Benders decomposition, Lagrangian relaxation or relax-and-cut. The considered problems include the (prize-collecting) Steiner tree problem [17], [33], tree-star problems [32], the shared arborescence problem [34], the upgrading spanning tree problem [36] and for maximum-weight connected subgraph problems [35].

We also studied models arising in the design of switched Ethernet networks implementing the Multiple Spanning Tree Protocol [24]. In these problems, multiple spanning trees have to be established in a network to route demands partitioned into virtual local access networks. Different mixed-integer formulations for the problem have been proposed and compared, both theoretically and computationally.

**Dynamic programming for the minimum-cost maximal knapsack packing problem:**
Given a set of items with profits and weights and a knapsack capacity, we studied the problem of finding a maximal knapsack packing that minimizes the profit of the selected items. We proposed an effective dynamic programming (DP) algorithm which has a pseudo-polynomial time complexity. We demonstrated the equivalence between this problem and the problem of finding a minimal knapsack cover that maximizes the profit of the selected items. In an extensive computational study on a large and diverse set of benchmark instances, we demonstrated that the new DP algorithm outperforms a state-of-the-art commercial mixed-integer programming (MIP) solver applied to the two best performing MIP models from the literature [25].