New Insights on Differential and Linear Bounds Using Mixed Integer Linear Programming

Current Mixed Integer Linear Programming (MILP)-based search against symmetric-key primitives with 8-bit S-boxes can only build word-wise model to search for truncated differential characteristics. In such a model, the properties of the Differential Distribution Table (DDT) are not considered. To take these properties into account, a bit-wise model is necessary, which can be generated by the H-representation of the convex hull or the logical condition modeling. However, the complexity of both approaches becomes impractical when the size of the S-box exceeds 5 bits. In this paper, we propose a new modeling for large (8-bit or more) S-boxes. In particular, we first propose an algorithm to generate a bit-wise model of the DDT for large S-boxes. We observe that the problem of generating constraints in logical condition modeling can be converted into the problem of minimizing the product-of-sum of Boolean functions, which is a well-studied problem. Hence, classical off-the-shelf solutions such as the Quine-McCluskey algorithm or the Espresso algorithm can be utilized, which makes building a bit-wise model, for 8-bit or larger S-boxes, practical. Then this model is further extended to search for the best differential characteristic by considering the probabilities of each propagation in the DDT, which is a much harder problem than searching for the lower bound on the number of active S-boxes. Our idea is to separate the DDT into multiple tables for each probability and add conditional constraints to control the behavior of these multiple tables. The proposed modeling is first applied to SKINNY-128 to find that there is no differential characteristic having probability higher than 2 −128 for 14 rounds, while the designers originally expected that 15 rounds were required. We also applied the proposed modeling to two, arbitrarily selected, constructions of the seven AES round function based constructions proposed in FSE 2016 and managed to improve the lower bound on the number of the active S-boxes in one construction and the upper bound on the differential characteristic for the other.

Supply chain network design and operation: Systematic decision-making for centralized, distributed, and mobile biofuel production using mixed integer linear programming (MILP) under uncertainty

This article presents a planning problem in a distribution network incorporating two levels inventory management of perishable products, lot-sizing, multi-sourcing and transport capacity with a homogeneous fleet of vehicles. A mixed integer linear programming (MILP) and a greedy heuristic are developed to solve this real planning problem. There are some instances for which the solver cannot give a good lower bound within the limited time and for other instances it takes a lot of time to solve MILP. The greedy heuristic is an alternative to the mixed integer linear program to quickly solve some large instances taking into account original and difficult constraints. For some instances the gap between the solutions of the solver (MILP) and the heuristic becomes quite significant. The variable neighborhood descent (VND), the iterated local search (ILS) and the multi-start iterated local search (MS-ILS) are implemented. These methods are included in an APS (Advanced Planning System) and compared with a MILP solver. The instances are derived from actual data or built using a random generator of instances to have wider diversity for computational evaluation. The VND significantly improves the quality of solutions.

A computational comparison of cargo prioritization and terminal allocation problem models

Advances in sensors and cameras allow current research in convex hull algorithms to focus on defining methods capable of processing a big set of points. Typically, in most of these algorithms, the orientation function needs around five sums and two multiplications. In this paper, we propose SymmetricHull, a novel algorithm that, unlike the related ones, only performs two comparisons per point, discarding points with a low probability of belonging to the convex hull. Our algorithm takes advantage of the symmetric geometry of convex hulls in 2D spaces and relies on the convexity principle to get convex hulls, without needing further calculations. Our experiments show that SymmetricHull achieves good results, in terms of time and number of necessary operations, resulting especially efficient with sets of points between \(10^4\) and \(10^7\). Given that our datasets are organized by quadrants, the features of our algorithm can be summarized as follows: (1) a fast point discard based on known points with a good chance to be part of the convex hull, (2) a lexicographic sort of points with a high probability of belonging to the convex hull, and (3) a simple slope analysis to verify whether a point is within the convex hull or not.

This paper presents a novel approach to the joint optimization of job scheduling and data allocation in grid computing environments. We formulate this joint optimization problem as a mixed integer quadratically constrained program. To tackle the nonlinearity in the constraint, we alternatively fix a subset of decision variables and optimize the remaining ones via Mixed Integer Linear Programming (MILP). We solve the MILP problem at each iteration via an off-the-shelf MILP solver. Our experimental results show that our method significantly outperforms existing heuristic methods, employing either independent optimization or joint optimization strategies. We have also verified the generalization ability of our method over grid environments with various sizes and its high robustness to the algorithm setting.

The goal of this paper is to develop models and methods that use complementary strengths of Mixed Integer Linear Programming (MILP) and Constraint Programming (CP) techniques to solve problems that are otherwise intractable if solved using either of the two methods. The class of problems considered in this paper have the characteristic that only a subset of the binary variables have non-zero objective function coefficients if modeled as an MILP. This class of problems is formulated as a hybrid MILP/CP model that involves some of the MILP constraints, a reduced set of the CP constraints, and equivalence relations between the MILP and the CP variables. An MILP/CP based decomposition method and an LP/CP-based branch-and-bound algorithm are proposed to solve these hybrid models. Both these algorithms rely on the same relaxed MILP and feasibility CP problems. An application example is considered in which the least-cost schedule has to be derived for processing a set of orders with release and due dates using a set of dissimilar parallel machines. It is shown that this problem can be modeled as an MILP, a CP, a combined MILP-CP OPL model (Van Hentenryck 1999), and a hybrid MILP/CP model. The computational performance of these models for several sets shows that the hybrid MILP/CP model can achieve two to three orders of magnitude reduction in CPU time.

3-Points Convex Hull Matching (3PCHM) for fast and robust point set registration

We study the reduction of bounded reachability analysis of timed automata (TA) to a Mixed Integer Linear Programming (MILP) problem. While bounded model checking of timed automata has been explored in the literature based on the satisfiability of Boolean constraint formulas over linear arithmetic constraints verified using SAT Modulo Theory (SMT) solvers, the approach presented in this paper opens up the alternative of using MILP solvers. We present some preliminary results comparing the two approaches and provide ideas on how linear optimization can be useful for analyzing the behavior of TA. The results are supported by a prototype implementation which relies either on a MILP solver (Gurobi) or an SMT solver (MathSAT). Certain techniques for reducing the search space and improving the performance are also discussed.

Linear programming (LP) and mixed integer linear programming (MILP) problems belong among very important class of problems that find their applications in various managerial consequences. The aim of the paper is to discuss computational performance of current optimization packages for solving large scale LP and MILP optimization problems. Current market with LP and MILP solvers is quite extensive. Probably among the most powerful solvers GUROBI 6.0, IBM ILOG CPLEX 12.6.1, and XPRESS Optimizer 27.01 belong. Their attractiveness for academic research is given, except their computational performance, by their free availability for academic purposes. The solvers are tested on the set of selected problems from MIPLIB 2010 library that contains 361 test instances of different hardness (easy, hard, and not solved).

This article presents a planning problem in a distribution network incorporating two levels of inventory management. Perishable products are routed with lot-sizing, multi-sourcing and limited transport capacity using a homogeneous fleet of vehicles. A mixed integer linear programming (MILP) and a greedy heuristic have been developed to solve this real planning problem. There are some instances for which the solver cannot give a good lower bound within the limited time and for other instances it takes a lot of time to solve MILP. The greedy heuristic is an alternative to the mixed integer linear program, used to quickly solve some large instances taking into account original and difficult constraints. For some instances the gap between the solution provided by the solver (MILP) and the heuristic becomes quite significant. An iterated local search (ILS) using the variable neighborhood descent (VND) method has been implemented to improve the quality of heuristic solutions. We have included the ILS method in an APS (Advanced Planning System) and have compared it with an exact resolution of the MILP. Two types of instances are tested: models derived from actual data and models built using a random generator of instances that have wider diversity for computational evaluation. The ILS procedure significantly improves the quality of solutions and average computational time is much shorter than MILP resolution.

Distribution systems (DS) are a focal connection point for existing and new renewable energy sources. DS have increasing sizes; they connect increasing numbers of renewable energy sources and have a mandate to operate as financial entities seeking to maximize their profits in an electricity market. To achieve these goals, DS require highly efficient planning tools. In this paper, we propose a successive fuzzy mixed integer linear programming (MILP) method for optimally placing and sizing capacitors in a DS that explicitly models induction run wind electric generators (WEGs). The optimization problem is linearized, transformed into a fuzzy MILP model and is solved successively while updating the solution vector. The fuzzy MILP method employs fuzzy models of voltage constraints that are often violated in a planned system that lacks reactive power support. The use of fuzzy constraints makes the proposed solution method very robust by avoiding infeasibility and reducing the need for user expertise. Further, the fuzzy MILP method represents the dual objectives of capacitor cost and annual energy loss minimization as fuzzy sets with individual satisfaction functions. The method maximizes the minimum of these satisfaction functions during optimization. Tests on 69-bus and 2002-bus DS show that the method performs well and is robust.

Draw management is a critical part of the successful recovery of mineral reserves by cave mining. This thesis presents a draw control model that indirectly increases resource value by controlling production based on geotechnical constraints. The mixed integer linear programming (MILP) model is formulated as a goal programming model that includes seven general constraint types. These constraints model the mining system and drive the operation towards the dual strategic targets of total monthly production tonnage and cave shape. This approach increases value by ensuring that reserves are not lost due to poor draw practice. The model also allows any number of processing plants to feed from multiple sources (caves, stockpiles, and dumps). The ability to blend material allows the model to be included in strategic level studies that target corporate objectives while emphasising production control within each cave. There are three main production control constraints in the MILP. The first of these, the draw maturity rules, is designed to balance drawpoint production with cave propagation rates. The maturity rules are modelled using disjunctive constraints. The constraint regulates production based on drawpoint depletion. Drawpoint production increases from 100 mm/d to 404 mm/d once the drawpoint reaches 6.5% depletion. Draw can continue at this maximum rate until drawpoint ramp-down begins as 93.5% depletion. The maximum draw rate decreases to 100 mm/d at drawpoint closure in the three maturity rule systems included in the thesis. The maturity rule constraints combine with the minimum draw rate constraint to limit production based on the difference between the actual and ideal drawpoint depletion. Drawpoints which lag behind their ideal depletion are restricted by the maturity rules while those that exceeded the ideal depletion were forced to mine at their minimum rate to ensure that cave porosity was maintained. The third production control constraint, relative draw rate (RDR), prohibits isolated draw by ensuring that extraction is uniform across the cave. It does this by controlling the relative draw difference between adjacent drawpoints. It is apparent in this thesis that production from a drawpoint can have an indirect effect on remote drawpoints because the relative draw rate constraints pass from one neighbour to the next within the cave. Tightening the RDR constraint increases production variation during cave ramp-up. This variation occurs because the maturity rules dictate that new drawpoints must produce at a lower draw rate than mature drawpoints. As a result, newly opened drawpoints limit production from the mature drawpoints within their region of the cave (not just their immediate neighbours). The MILP is also used to quantify production changes caused by varying geotechnical constraints, limiting haulage capacity, and reversing mining direction. It has been shown that tightening the RDR constraint decreases total cave production. The ramp-up duration also increased by eighteen months compared to the control RDR scenario. Tighter relative draw also made it difficult to maintain cave shape during ramp-up. However, once ramp-up was complete, the tighter control produced a better depletion surface. The trial with limited haulage capacity identified bottlenecks in the materials handling system. The main bottlenecks occur in the production drives with the greatest tonnage associated with their drawpoints. There also appears to be an average haulage capacity threshold for the extraction drives of 2000 tonnes per drawpoint. Only one drive with a capacity below this threshold achieves its target production in each period. Reversing the cave advance to initiate in the South-East shows the greatest potential for achieving total production and cave shape targets. The greater number of drawpoints available early in the schedule provides more production capacity. This ability to distribute production over a greater number of drawpoints reduces the total production lag during ramp-up. In addition to its role in feasibility studies, the MILP is well suited for use as a production guidance tool. It has been shown in three case studies that the model can be used to evaluate production performance and to establish long term production targets. The first of the studies shows the analysis of historical production data by comparison to the MILP optimised schedule. The second shows that the model produces an optimised production plan irrespective of the current cave state. The final case study emulates the draw control cycle used by the Premier Diamond Mine. The series of optimised production schedules mirror that of the life-of-mine schedule generated at the start of the iterative process. The results illustrate how the MILP can be used by a draw control engineer to analyse production data and to develop long term production targets both before and after a cave is brought into full production.

We present a general approach to the FPGA technology mapping problem that applies to any logic block composed of lookup tables (LUTs) and can yield optimal solutions. The connections between LUTs of a logic block are modeled by virtual switches, which define a set of multiple-LUT blocks (MLBs) called an MLB-basis. We identify the MLB-bases for various commercial logic blocks. Given a n MLB-basis, we formulate FPGA mapping as a mixed integer linear programming (MILP) problem to achieve both the generality and the optimality objectives. We solve the MILP models using a general-purpose MILP solver, and present the results of mapping some ISCAS.85 benchmark circuits with a variety of commercial FPGAs. Circuits of a few hundred gates can be mapped in reasonable time using the MILP approach directly. Larger circuits can be handled by partitioning them prior to technology mapping. We show that optimal or provably near-optimal solutions can be obtained for the large ISCAS.85 benchmark circuits using partitions defined by their high-level functions.

Two new continuous-time MILP (Mixed Integer Linear Programming) models are developed for resource constrained project scheduling problems in which activities can be conducted in different (multi) modes. There are efficient and flexible MILP models for this problem in which time is represented by a number of discrete time intervals. However, they require to use large numbers of binary variables as the number of activities increases and time horizon stretches. Hence efforts have been made to formulate MILP models based on continuous-time representation and therefore to speed up the solution process. We propose two novel MILP models based on the event and resource flow concepts. The models are verified by solving benchmark problems in the literature.