Vishnu Narayanan

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.

Polymatroids and mean-risk minimization in discrete optimization

We study discrete optimization problems with a submodular mean-risk minimization objective. For 0-1 problems a linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic valid inequalities. We report computational experiments on risk-averse capital budgeting problems with uncertain returns.

The submodular knapsack polytope

The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various assumptions, a probabilistic constraint on 0-1 variables can be modeled as a submodular knapsack set. In this paper we describe cover inequalities for the submodular knapsack set and investigate their lifting problem. Each lifting problem is itself an optimization problem over a submodular knapsack set. We give sequence-independent upper and lower bounds on the valid lifting coefficients and show that whereas the upper bound can be computed in polynomial time, the lower bound problem is NP-hard. Furthermore, we present polynomial algorithms based on parametric linear programming and computational results for the conic quadratic 0-1 knapsack case.

Cuts for Conic Mixed-Integer Programming

A conic integer program is an integer programming problem with conic constraints. Conic integer programming has important applications in finance, engineering, statistical learning, and probabilistic integer programming. Here we study mixed-integer sets defined by second-order conic constraints. We describe general-purpose conic mixed-integer rounding cuts based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve continuous conic programming relaxations at the nodes of the search tree. Our preliminary computational experiments with the new cuts show that they are quite effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs.

Exact approaches for static data segment allocation problem in an information network

In a large distributed database, data are geographically distributed across several separate servers (or data centers). This helps in distributing load in the access network. It also helps to serve data locally where it is required. There are various approaches based on the granularity of data for efficient data distribution in a communication network. The file allocation problem (FAP) locates files to servers, the segment allocation problem (SAP) locates database segments, and the mirror location problem (MLP) locates replicas of the entire database. The placement of such data to multiple servers can be modeled as an optimization problem. The major decisions influencing optimization involves the location of servers, allocation of content and assignment of users. In this paper, we study the segment allocation problem (SAP), which is also known as the partial mirroring problem. This approach is more tractable than the file allocation problem in realistic cases and also eliminates the overhead of (constant) update costs that is incurred in the mirror location problem. Our contribution is two-fold: Firstly, earlier works on SAP assume pre-defined segments. We build a data partitioning method using well-known facility location models. We quantify the performance of the partitioning method. We show that the method partitions the database within a reasonable limit of error. Secondly, we introduce a new model for the segment allocation problem in which the segments are completely connected to each other by high-bandwidth links and contains a cost benefit for inter-segment traffic flows. We formulate this problem as an MILP and build exact solution approaches to solve large scale problems. We demonstrate some structural properties of the problem that make it solvable, using a Benders decomposition algorithm. Computational results validate the superiority of the decomposition approach.

Mathematical models and empirical analysis of a simulated annealing approach for two variants of the static data segment allocation problem

We consider a content distribution network (CDN) in which data hubs or servers are established in multiple locations to cater to local demands. The distributions of data to these hubs along with related network design problems (such as hub location and user assignment) are the key decision problems to consider to minimize the total routing cost. A new model for allocation of segments is introduced in Sen, Krishnamoorthy, Rangaraj and Narayanan, Comput Oper 62 (2015), 282–295, in which local preferences guide the database partitioning process, and the servers are fully connected to each other. In this article, we develop a simulated annealing (SA) approach (referred to as SA-mesh) to solve this problem and compare its performance with the corresponding mixed-integer linear programming (MILP) formulation. We also formulate a much harder variant of the problem in which servers are interconnected by a tree. We develop a SA algorithm (referred to as SA-tree) for this variant, in which a local search is incorporated to find a suboptimal tree backbone. We use a customized data structure based on linked lists to represent a solution in our algorithms. This enables our algorithms to scale to much larger instances of the problem. We use optimal solutions and the benchmarks obtained by CPLEX to justify the performance of our algorithms.

Facility location models to locate data in information networks: a literature review

The usage of the Internet has grown substantially in recent times. This has resulted in high volumes of data traffic. There is a concomitant rise in bandwidth demands that could result in excessive download delays (or latency). Thus, a single-server system is no more a prudent choice for data storage. Replication of content and placing them on multiple servers is a method that is used to reduce latency. However, this solution comes at a huge cost. Moreover, replicating objects randomly does not necessarily improve system performance. It is possible to arrive at a solution to the problem of placing content so as to achieve better cost performance. Other performance measures include latency, load balancing and data availability. We refer to the problem of locating content as data location problem in information networks, or DLPIN. The choice of server locations, query routing strategy and user assignment are some of the important problems that require attention along with the location of the data/content. Resource constraints and the nature of traffic (static/dynamic) are two important parameters in the problem environment, and therefore are key distinguishing features in the models. The main contribution of this paper is a novel classification and study of DLPIN on the basis of problem features. The research in this area started with files, the smallest units of allocation. Gradually, files and programs, database segments and entire databases (or mirrors) have been studied. We design examples from these use cases to elaborate a variety of problems in a comprehensive review. Facility location models from physical logistics are extensively used to model these problems. Our paper presents a literature survey of such mathematical models for data location problems. We present a gap analysis that provides pointers to possible future research in this area. This paper also serves to document the success in the use of mathematical programming approaches for data location in information networks.

History-dependent scheduling

In this paper, we extend job scheduling models to include aspects of history-dependent scheduling, where setup times for a job are affected by the aggregate activities of all predecessors of that job. Traditional approaches to machine scheduling typically address objectives and constraints that govern the relative sequence of jobs being executed using available resources. This paper optimises the operations of multiple unrelated resources to address sequential and history-dependent job scheduling constraints along with time window restrictions. We denote this consolidated problem as the general precedence scheduling problem (GPSP). We present several applications of the GPSP and show that many problems in the literature can be represented as special cases of history-dependent scheduling. We design new ways to model this class of problems and then proceed to formulate it as an integer program. We develop specialized algorithms to solve such problems. An extensive computational analysis over a diverse family of problem data instances demonstrates the efficacy of the novel approaches and algorithms introduced in this paper. HighlightsDetailed explanation of differences between GPSP and Block-world problem has been submitted within the response document.Summary of differences between GPSP and block-world problem has been included in the paper.Justification for using only first log(n) and not all i, j, K combinations has been included in the paper.Mistakes in write-up have been corrected as pointed out by reviewers.

Approaches for solving the container stacking problem with route distance minimization and stack rearrangement considerations

We consider an optimization problem of sequencing the operations of cranes that are used for internal movement of containers in maritime ports. Some features of this problem have been studied in the literature as the stacker crane problem (SCP). However, the scope of most literature (including SCP) is restricted to minimizing the route or distance traveled by cranes and the resulting movement-related costs. In practice, cargo containers are generally stacked or piled up in multiple separate columns, heaps or stacks at ports. So, the cranes need to often rearrange or shuffle such container stacks, in order to pick up any required container. If substantial re-stacking is involved, cranes expend considerable effort in container stack rearrangement operations. The problem of minimizing the total efforts/time of the crane must therefore account for both - the stack rearrangement costs and also the movement-related (route distance) costs. The consolidated problem differs from standard route distance minimization situations if stack rearrangement activities are considered. We formally define the consolidated problem, identify its characteristic features and hence devise suitable models for it. We formulate several alternative MIP approaches to solve the problem. We compare the performance of our MIP formulations and analyze their suitability for various possible situations.

Intersection cuts for convex mixed integer programs from translated cones

We develop a general framework for linear intersection cuts for convex integer programs with full-dimensional feasible regions by studying integer points of their translated tangent cones, generalizing the idea of Balas (1971). For proper (i.e, full-dimensional, closed, convex, pointed) translated cones with fractional vertices, we show that under certain mild conditions all intersection cuts are indeed valid for the integer hull, and a large class of valid inequalities for the integer hull are intersection cuts, computable via polyhedral approximations. We also give necessary conditions for a class of valid inequalities to be tangent halfspaces of the integer hull of proper translated cones. We also show that valid inequalities for non-pointed regular translated cones can be derived as intersection cuts for associated proper translated cones under some mild assumptions.