Abstract: In an embodiment, a set of parameters associated with a vehicle routing problem is received. A set of decision variables and a set of constraints associated with an optimization problem are received. An optimization problem is constructed. The optimization problem is divided into a set of sub-problems. Each of the set of sub-problems corresponds to a subset of warehouses of a set of warehouses. An intermediate solution is determined for each of the set of sub-problems to determine a set of routes associated with the corresponding sub-problem of the set of sub-problems. The intermediate solution associated with each of the set of sub-problems is combined to determine a final solution of the optimization problem based on the received set of constraints. The determined final solution is indicative of the set of optimal routes to be assigned to the set of vehicles. The final solution is rendered on a display device.

Abstract: A method may include obtaining a set of tags and a set of items in which each item is pre-sorted into a cluster and each item corresponds to one or more tags. The method may include generating a bipartite graph that includes the set of tags as a first set of nodes and the clusters of items as a second set of nodes. Relationships between tags and items may be represented as edges between the first nodes and the second nodes. The bipartite graph may be modeled as a quadratic programming formulation, and cluster descriptor sets that each include one or more of the tags may be determined by solving the quadratic programming formulation of the bipartite graph, each of the cluster descriptor sets providing an explanation of how one or more clusters of items were pre-sorted. The method may include analyzing the items based on the luster descriptor sets.

Abstract: According to an aspect of an embodiment, operations include receiving a Quadratic Integer Programming (QIP) problem including an objective function and a set of constraints on integer variables associated with the objective function. The operations further include obtaining an approximation of the QIP problem by relaxing the QIP problem and generating an approximate solution by solving the obtained approximation. The operations further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation of the QIP problem based on the generated approximate solution and the received QIP problem. The operations further include submitting the generated QUBO formulation to an optimization solver machine and receiving a solution of the submitted QUBO formulation from the optimization solver machine. The operations further include publishing an integral solution of the received QIP problem on a user device based on the received solution.

Abstract: A method of solving a large scale combinatorial optimization problem including inputting, via at least one processor, an objective function and an initial solution as a mapping from a plurality of n nodes, randomly clustering the plurality of nodes into k clusters of n/k nodes each, for each cluster of the k clusters, assigning binary variables to denote each possible permutation of a label set within the cluster, determining that there are u=k2 variables if k>2, and u=1 variables if k=2, expressing the objective function in terms of the un/k variables, solving the objective function in terms of the un/k variables using a Quadratic Unconstrained Binary Optimization (QUBO) solver to obtain an updated solution, determining whether a convergence criteria is satisfied for the updated solution, and upon a determination that a convergence criteria is satisfied, outputting the updated solution to the objective function.

Abstract: A method may include assigning each node of a network to a single first node cluster and selecting nodes of the network as a first set of nodes. The method may further include solving an optimization problem by reassigning one or more of the nodes of the first set of nodes to a second node cluster while maintaining the nodes that are not part of the first set of nodes in the first node cluster. The method may also include after solving the optimization problem, selecting other nodes of the network as another set of nodes and resolving the optimization problem by reassigning one or more of the nodes of the other set of nodes to a third node cluster while maintaining the node cluster assignment of the nodes that are not part of the other set of nodes.

Abstract: According to an aspect of an embodiment, operations may include receiving a set of inputs associated with a set of orders, a set of production lines, and timelines for the production. The operations may further include initializing each of a set of intervals to be used for scheduling of the production, based on a first interval size. The operations may further include generating a first Quadratic Unconstrained Binary Optimization (QUBO) formulation. The operations may further include generating a first solution of the first QUBO formulation. The operations may further include updating each of the initialized set of intervals based on a second interval size. The operations may further include generating a second QUBO formulation. The operations may further include generating a second solution by solving the second QUBO formulation and determining a schedule to be used for the production of the set of orders, based on the generated second solution.

Abstract: A method may include obtaining a first matrix that represents data in a data set and obtaining a number of clusters into which the data is to be grouped. The method may further include constructing a second matrix using the first matrix and the number of clusters. The second matrix may represent a formulation of a first optimization problem in a framework of a second optimization problem. The method may further include solving the second optimization problem using the second matrix to generate a solution of the second optimization problem and mapping the solution of the second optimization problem into a first solution matrix that represents a solution of the first optimization problem. The method may further include grouping the data into multiple data clusters using the first solution matrix. A number of the multiple data clusters may be equal to the number of clusters.

Abstract: According to an aspect of an embodiment, operations include receiving an Integer Linear Programming (ILP) problem including an objective function and a set of constraints on integer variables of the objective function. The operations may further include determining a lower bound vector for the integer variables and determining an upper bound vector for the integer variables. The operations further include obtaining a binary variable representation of each of the integer variables and updating the received ILP problem based on the obtained binary variable representation. The operations further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation of the updated ILP problem and submitting the generated QUBO formulation to a first optimization solver machine. The operations further include receiving a solution of the submitted QUBO formulation and determining an integral solution of the received ILP problem.

Abstract: According to an aspect of an embodiment, operations include obtaining a first set of input parameters associated with a modified multi-dimensional knapsack problem. The operations further include determining a lattice representation of the modified multi-dimensional knapsack problem and computing a reduced basis by applying a lattice reduction method on the set of basis vectors of the determined lattice representation. The operations further include determining a first mathematical formulation of the modified multi-dimensional knapsack problem based on the reduced basis and submitting the determined first mathematical formulation as an input to an integer linear programming (ILP) solver. The operations further include receiving a first solution of the submitted mathematical formulation from the ILP solver and determining an integral solution of the modified multi-dimensional knapsack problem. The operations further include controlling a user device to output the determined integral solution.

Abstract: A method may include assigning each node of a network to a single first node cluster and selecting nodes of the network as a first set of nodes. The method may further include solving an optimization problem by reassigning one or more of the nodes of the first set of nodes to a second node cluster while maintaining the nodes that are not part of the first set of nodes in the first node cluster. The method may also include after solving the optimization problem, selecting other nodes of the network as another set of nodes and resolving the optimization problem by reassigning one or more of the nodes of the other set of nodes to a third node cluster while maintaining the node cluster assignment of the nodes that are not part of the other set of nodes.

Abstract: According to an aspect of an embodiment, operations may include receiving a first input associated with a set of orders to be produced at a production facility and receiving a second input associated with a set of production lines. The operations may further include extracting a set of production-related datapoints and receiving a third input associated with a set of constraints. The operations may further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation based on the extracted set of datapoints and the third input and submitting the generated QUBO formulation to a first optimization solver machine. The operations may further include receiving a first solution of the submitted QUBO formulation from the first optimization solver machine and determining a schedule to be used for the production of the set of orders on the set of production lines, based on the received first solution.

Abstract: According to an aspect of an embodiment, operations include receiving a set of datapoints for integrated clustering and outlier detection. The operations further include receiving, as a first input, a clustering constraint comprising a number of outlier datapoints to be detected from the set of datapoints and a second input including a distance metric. The operations further include formulating an objective function based on the first and second inputs and transforming the objective function into an unconstrained binary optimization formulation. The operations further include providing such formulation as input to an optimization solver machine and generating a clustering result and an outlier detection result based on output of the optimization solver machine for the input. The clustering result includes a set of datapoint clusters, and the outlier detection result includes a set of outlier datapoints. The clustering result and the outlier detection result are published on a publisher system.

Abstract: A method may include obtaining a first matrix that represents data in a data set and obtaining a number of clusters into which the data is to be grouped. The method may further include constructing a second matrix using the first matrix and the number of clusters. The second matrix may represent a formulation of a first optimization problem in a framework of a second optimization problem. The method may further include solving the second optimization problem using the second matrix to generate a solution of the second optimization problem and mapping the solution of the second optimization problem into a first solution matrix that represents a solution of the first optimization problem. The method may further include grouping the data into multiple data clusters using the first solution matrix. A number of the multiple data clusters may be equal to the number of clusters.

Abstract: According to an aspect of an embodiment, operations include receiving a layout of a maritime facility and a first input including a vehicle count associated with a fleet of transport vehicles on the maritime facility. The operations further include determining a weighted graph representation of the maritime facility based on the received layout and generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation based on the weighted graph representation and the received first input. The operations further include submitting the generated QUBO formulation to a first optimization solver machine and receiving a first solution of the submitted QUBO formulation. The operations further include determining a set of paths to be traversed by the fleet of transport vehicles on the maritime facility for transporting the plurality of shipping containers to respective destination locations based on the received first solution.

Abstract: According to an aspect of an embodiment, operations include receiving a Quadratic Integer Programming (QIP) problem including an objective function and a set of constraints on integer variables associated with the objective function. The operations further include obtaining an approximation of the QIP problem by relaxing the QIP problem and generating an approximate solution by solving the obtained approximation. The operations further include generating a Quadratic Unconstrained Binary Optimization (QUBO) formulation of the QIP problem based on the generated approximate solution and the received QIP problem. The operations further include submitting the generated QUBO formulation to an optimization solver machine and receiving a solution of the submitted QUBO formulation from the optimization solver machine. The operations further include publishing an integral solution of the received QIP problem on a user device based on the received solution.

Abstract: A method may include assigning each node of a network to a different node cluster such that a number of nodes equals a number of node clusters and selecting multiple of the nodes of the network as a set of nodes. The method may further include solving a first optimization problem by reassigning one or more of the nodes of the set of nodes to a different node cluster while maintaining assigned node clusters of the nodes that are not part of the set of nodes and after solving the first optimization problem, selecting multiple of the node clusters as a set of node clusters. The method may also include solving a second optimization problem by merging two or more of the node clusters of the set of node clusters while maintaining the node clusters that are not part of the set of node clusters.

Abstract: A method may include assigning each node of a first network to a different node cluster such that a number of nodes equals a number of node clusters, selecting multiple nodes of the first network as a set of nodes, and selecting multiple node clusters as a set of node clusters. The method may also include solving a first optimization problem by reassigning one or more of the nodes of the set of nodes to different node clusters of the set of node clusters while maintaining assigned node clusters of the nodes that are not part of the set of nodes and after reassigning one or more of the nodes of the set of nodes to different node clusters, merging the nodes assigned to at least one of the node clusters to form a second network with fewer nodes than the number of nodes of the first network.

Abstract: According to an aspect of an embodiment, operations include receiving a set of datapoints for integrated clustering and outlier detection. The operations further include receiving, as a first input, a clustering constraint comprising a number of outlier datapoints to be detected from the set of datapoints and a second input including a distance metric. The operations further include formulating an objective function based on the first and second inputs and transforming the objective function into an unconstrained binary optimization formulation. The operations further include providing such formulation as input to an optimization solver machine and generating a clustering result and an outlier detection result based on output of the optimization solver machine for the input. The clustering result includes a set of datapoint clusters, and the outlier detection result includes a set of outlier datapoints. The clustering result and the outlier detection result are published on a publisher system.

Abstract: A method of solving a large scale combinatorial optimization problem including inputting, via at least one processor, an objective function and an initial solution as a mapping from a plurality of n nodes, randomly clustering the plurality of nodes into k clusters of n/k nodes each, for each cluster of the k clusters, assigning binary variables to denote each possible permutation of a label set within the cluster, determining that there are u=k2 variables if k>2, and u=1 variables if k=2, expressing the objective function in terms of the un/k variables, solving the objective function in terms of the un/k variables using a Quadratic Unconstrained Binary Optimization (QUBO) solver to obtain an updated solution, determining whether a convergence criteria is satisfied for the updated solution, and upon a determination that a convergence criteria is satisfied, outputting the updated solution to the objective function.

Abstract: A method of converting a HOBO problem into a QUBO problem. The method may include creating a data structure of key-value pairs by sorting the plurality of indices of the variables of the HOBO problem, the key in each key-value pair corresponding to combinations of quadratic terms appearing in the HOBO and the value corresponding to all terms of at least degree three that contain the associated key. For each key of the data structure, a quadratization process is performed including identifying a key with the largest number of associated values, replacing the identified key with an auxiliary variable, and updating the data structure so as to correspond with the replacement of the auxiliary variable, storing the auxiliary variable and a quadratic term the auxiliary variable replaced as a pair in a data map. The method may also include constructing a quadratic polynomial for each pair in the data map.