Conic Quadratic Programming Research Articles

The Jordan algebraic structure of the rotated quadratic cone

In this paper, we look into the rotated quadratic cone and analyze its algebraic structure. We construct an algebra associated with this cone and show that this algebra is a Euclidean Jordan algebra (EJA) with a certain inner product. We also demonstrate some spectral and algebraic characteristics of this EJA. The rotated quadratic cone is then proven to be the cone of squares of the generated EJA. The obtained results can help optimization researchers improve specialized interior-point algorithms for rotated quadratic cone programming based on the generated EJA. Additionally, since it is known that the rotated quadratic cone is a special case of the power cone, another reason for this study may be to open the door to understanding the algebraic structure of the general power cone in the future.

Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications

A Unifying Framework for the Convexification of Mixed-Integer Conic Binary Sets The paper “Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications,” by Fatma Kilinc-Karzan, Simge Kucukyavuz, Dabeen Lee, and Soroosh Shafieezadeh-Abadeh, develops a unifying framework for convexifying mixed-integer conic binary sets. Many applications in machine-learning and operations research give rise to integer programming models with nonlinear structures and binary variables. The paper develops general methods for generating strong valid inequalities that take into account multiple conic constraints at the same time. The authors demonstrate that their framework applies to conic quadratic programming with binary variables, fractional programming, best subset selection, distributionally robust optimization, and sparse approximation of positive semidefinite matrices.

Joint location and cost planning in maximum capture facility location under random utilities

We study a joint facility location and cost planning problem in a competitive market under random utility maximization (RUM) models. The objective is to locate new facilities and make decisions on the costs (or budgets) to spend on the new facilities, aiming to maximize an expected captured customer demand, assuming that customers choose a facility among all available facilities according to a RUM model. We examine two RUM frameworks in the discrete choice literature, namely, the additive and multiplicative RUM. While the former has been widely used in facility location problems, we are the first to explore the latter in the context. We numerically show that the two RUM frameworks can well approximate each other in the context of the cost optimization problem. In addition, we show that, under the additive RUM framework, the resultant cost optimization problem becomes highly non-convex and may have several local optima. In contrast, the use of the multiplicative RUM brings several advantages to the competitive facility location problem. For instance, the cost optimization problem under the multiplicative RUM can be solved efficiently by a general convex optimization solver, or can be reformulated as a conic quadratic program and handled by a conic solver available in some off-the-shelf solvers such as CPLEX or GUROBI. Furthermore, we consider a joint location and cost optimization problem under the multiplicative RUM and propose three approaches to solve the problem, namely, an equivalent conic reformulation, a multi-cut outer-approximation algorithm, and a local search heuristic. We provide numerical experiments based on synthetic instances of various sizes to evaluate the performances of the proposed algorithms in solving the cost optimization and the joint location and cost optimization problems.

Shipment planning and safety stock placement in maritime supply chains with stochastic demand and transportation times

This paper introduces and studies shipment planning and safety stock placement in bulk and semi-bulk maritime supply chains. This problem jointly optimizes the assignment of destinations to origins, shipment quantities of multiple products by determining the share of the contract by sea transport options, sailing speed, and multi-echelon safety stocks with the objective to minimize the expected total transportation and inventory cost. In the case of a serial system supplying a single product, we analytically characterize the effect of sailing speed and different time and cost parameters on optimal safety stock placement. For the general case, the integrated problem can be solved in four different ways. We present a mixed integer nonlinear program (MINLP) and the superior reformulation of the mixed integer conic quadratic integer program (MICQP) that can be solved using state-of-the-art solvers. Then, we propose two exact solution procedures that iteratively approximate the nonlinear terms in the objective function and constraints. Our numerical experiments illustrate the interdependence between safety stock placement, assignment decisions, and sailing speed. Results also show that it is important to adopt multi-stage inventory optimization and there is an added value in integrating shipment planning and safety stock placement. Furthermore, computational experiments demonstrate that all the solution approaches are alternatives to solving small instance sizes. For medium and large instances, the solvers are unable to provide quality solutions within one hour while the iterative procedures find optimal or near-optimal solutions in reasonable CPU times.

Constrained multi‐location assortment optimization under the multinomial logit model

AbstractWe study the assortment optimization problem in an online setting where a retailer uses multiple distribution centers (DC) to fulfill orders from multiple regions. Customer choice in each region follows a multinomial logit model. Each DC can carry up to a pre‐specified number of products. Outbound shipping cost to a region depends on the DC that ships the order. The problem is to determine which products to carry in each DC and which products to offer for sale in each region to maximize the expected profit. We first show that the problem is NP‐complete. We develop a conic quadratic mixed integer programming formulation and suggest a family of valid inequalities. We also show that a special case with identical choice models can be solved as a linear program. This LP solution approach can be used to develop heuristics for the general case. Numerical experiments show that our conic approach outperforms the mixed integer linear programming formulation and enables us to solve moderately sized instances optimally. The experiments also show that not allowing cross‐shipments or not considering them in assortment decisions may lead to substantial losses and LP‐based heuristics can be effective in practice.

An Algebraic-Based Primal–Dual Interior-Point Algorithm for Rotated Quadratic Cone Optimization

In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

Gradient boosting for convex cone predict and optimize problems

Prediction models are traditionally optimized independently from decision-based optimization. Conversely, a ‘smart predict then optimize’ (SPO) framework optimizes prediction models to minimize downstream decision regret. In this paper we present dboost, the first general purpose implementation of smart gradient boosting for ‘predict, then optimize’ problems. The framework supports convex quadratic cone programming and gradient boosting is performed by implicit differentiation of a custom fixed-point mapping. Experiments comparing with state-of-the-art SPO methods show that dboost can further reduce out-of-sample decision regret.

A generalized Benders decomposition approach for the mean-standard deviation shortest path problem

ABSTRACT This paper concentrates on the mean-standard deviation shortest path problem, which is an important extension of traditional shortest path problem. Due to the standard deviation term, the general formulation of this problem is nonlinear and concave. We transform this formulation into a mixed-integer conic quadratic program and develop a generalized Benders decomposition approach. The Benders master problem is a continuous conic quadratic program about travel time mean and standard deviation. The subproblem is a least expected travel time path problem with the variance limit. At each iteration, the subproblem generates a generalized Benders optimality cut for the relaxed Benders master problem. The relaxed Benders master problem provides an ascending lower bound and the subproblem produces a feasible solution to update the upper bound. In the numerical experiments, all instances in four transportation networks are solved optimally. This paper provides a novel solving scheme for the mean-standard deviation shortest path problem.

Optimal vault problem – form finding through 2D convex program

This work puts forward a form finding problem of designing a least-volume vault that is a surface structure spanning over a plane region, which via pure compression transfers a vertically tracking load to the supporting boundary. Through a duality scheme, developed recently for the design of pre-stressed membranes, the optimal vault problem is reduced to a pair of mutually dual convex problems formulated on the 2D reference region. The vault constructed upon solutions of those problems is both of minimum volume and minimum compliance; analytical examples of optimal vaults are given. Through a measure-theoretic approach, such optimal vaults are proved to solve the Prager problem of designing a 3D structure that by compression carries a transmissible load. The ground structure method applied to the convex problems furnishes a pair of discrete, conic quadratic programs leading to optimal design of grid-shells. By adopting the member-adding adaptive technique, this pair is efficiently tackled numerically. The computer method is demonstrated on a number of examples where highly precise grid-shell approximations of optimal vaults are found.

Multi-objective regression modeling for natural gas prediction with ridge regression and CMARS

Residential customers are the main users generally need a great quantity of natural gas in distribution systems, especially, in the wintry weather season since it is particularly consumed for cooking and space heating. Hence, it ought to be non-interruptible. Since distribution systems have a restricted ability for supply, reasonable planning and prediction through the whole year, especially in winter seasons, have emerged as vital. The Ridge Regression (RR) is formulated mainly to decrease collinearity results through shrinking the regression coefficients and reducing the impact in the model of variables. Conic multivariate adaptive regression splines ((C)MARS) model is constructed as an effective choice for MARS by using inverse problems, statistical learning, and multi-objective optimization theories. In this approach, the model complexity is penalized in the structure of RR and it is constructed a relaxation by utilizing continuous optimization, called Conic Quadratic Programming (CQP). In this study, CMARS and RR are applied to obtain forecasts of residential natural gas demand for local distribution companies (LDCs) that require short-term forecasts, and the model performances are compared by using some criteria. Here, our analysis shows that CMARS models outperform RR models. For one-day-ahead forecasts, CMARS yields a MAPE of about 4.8%, while the same value under RR reaches 8.5%. As the forecast horizon increases, it can be seen that the performance of the methods becomes worse, and for a forecast one week ahead, the MAPE values for CMARS and RR are 9.9% and 18.3%, respectively.

Operator Splitting for a Homogeneous Embedding of the Linear Complementarity Problem

We present a first-order quadratic cone programming algorithm that can scale to very large problem sizes and produce modest accuracy solutions quickly. Our algorithm returns primal-dual optimal solutions when available or certificates of infeasibility otherwise. It is derived by applying Douglas--Rachford splitting to a homogeneous embedding of the linear complementarity problem, which is a general set membership problem that includes quadratic cone programs (QCPs) as a special case. Each iteration of our procedure requires projecting onto a convex cone and solving a linear system with a fixed coefficient matrix. If a sequence of related problems are solved, then the procedure can easily be warm-started and make use of factorization caching of the linear system. We demonstrate on a range of public and synthetic datasets that for feasible problems our approach tends to be somewhat faster than applying operator splitting directly to the QCP, and in cases of infeasibility our approach can be significantly faster than alternative approaches based on diverging iterates. The algorithm we describe has been implemented in C and is available open-source in the solver SCS v3.0.

Estimation in the partially nonlinear model by continuous optimization

ABSTRACT A useful model for data analysis is the partially nonlinear model where response variable is represented as the sum of a nonparametric and a parametric component. In this study, we propose a new procedure for estimating the parameters in the partially nonlinear models. Therefore, we consider penalized profile nonlinear least square problem where nonparametric components are expressed as a B-spline basis function, and then estimation problem is expressed in terms of conic quadratic programming which is a continuous optimization problem and solved interior point method. An application study is conducted to evaluate the performance of the proposed method by considering some well-known performance measures. The results are compared against parametric nonlinear model.

Max-Min Fairness and Sum Throughput Maximization for In-Band Full-Duplex IoT Networks: User Grouping, Bandwidth and Power Allocation

The skyrocketing growth in the number of Internet of Things (IoT) devices has posed a huge traffic demand for fifth-generation (5G) wireless networks and beyond. In-band full-duplex (IBFD), which is theoretically expected to double the spectral efficiency of a half-duplex wireless channel and connect more devices, has been considered as a promising technology in order to accelerate the development of IoT. In order to exploit the full potential of IBFD, the key challenge is how to handle network interference (including self-interference, co-channel interference, and multiuser interference) more effectively. In this paper, we propose a simple yet efficient user grouping method, where a base station (BS) serves strong downlink users and weak uplink users and vice versa in different frequency bands, mitigating severe network interference. First, we aim to maximize a minimum rate among all of the users subject to bandwidth and power constraints, which is formulated as a nonconvex optimization problem. By leveraging the inner approximation framework, we develop a very efficient iterative algorithm for solving this problem, which guarantees at least a local optimal solution. The proposed iterative algorithm solves a simple convex program at each iteration, which can be further cast to a conic quadratic program. We then formulate the optimization problem of sum throughput maximization, which can be solved by the proposed algorithm after some slight modifications. Extensive numerical results are provided to show not only the benefit of using full-duplex radio at BS, but also the advantage of the proposed user grouping method.

Autonomous and conventional bus fleet optimization for fixed-route operations considering demand uncertainty

The emerging technology of autonomous vehicles has been widely recognized as a promising urban mobility solution in the future. This paper considers the integration of autonomous vehicles into bus transit systems and proposes a modeling framework to determine the optimal bus fleet size and its assignment onto multiple bus lines in a bus service network considering uncertain demand. The mixed-integer stochastic programming approach is applied to formulate the problem. We apply the sample average approximation (SAA) method to solve the formulated stochastic programming problem. To tackle the nonconvexity of the SAA problem, we first present a reformulation method that transforms the problem into a mixed-integer conic quadratic program (MICQP), which can be solved to its global optimal solution by using some existing solution methods. However, this MICQP based approach can only handle the small-size problems. For the cases with large problem size, we apply the approach of quadratic transform with linear alternating algorithm, which allows for efficient solution to large-scale instances with up to thousands of scenarios in a reasonable computational time. Numerical results demonstrate the benefits of introducing autonomous buses as they are flexible to be assigned across different bus service lines, especially when demand uncertainty is more significant. The introduction of autonomous buses would enable further reduction of the required fleets and total cost. The model formulation and solution methods proposed in this study can be used to provide bus transit operators with operational guidance on including autonomous buses into bus services, especially on the autonomous and conventional bus fleets composition and allocation.

Convergence of the direct limit analysis design method for discrete topology optimization

The present paper addresses the robustness and convergence behavior of the direct limit analysis (LA) based methodology developed for the topology design of continuum structures subject to prescribed statically and plastically admissible loads. The design methodology, based on a direct method formulation of the static LA problem, has recently been proposed for continuous topology optimization and its merits were highlighted. One of its remarkable features is the outstanding similarity of the topology design mathematical problem with its underlying direct static form of the LA problem. Subsequently, it has been extended to solve two dimensional discrete, i.e. black-and-white, topology design problems by modifying the objective function into a square root form in a way to penalize the intermediate densities and solving a sequence of conic quadratic programming problems of identical scale and algebraic structure as the continuous design problem, leaving a number of issues to be investigated, pertaining to convergence. In the present work different families of penalization function forms are proposed and assessed as alternatives to the original square root function. The performance is evaluated in terms of robustness, accuracy in the sense of closeness of the final design to a 0–1 topology and efficiency or number of approximate problems required for convergence. Convergence of the discrete topology design is shown to be improved using higher order power functions as well as trigonometric and exponential type penalty functions. The performance of the design method in solving example problems using the various penalization schemes is compared and the factors that affect it are analyzed.

A single allocation hub location and pricing problem

This study introduces a new problem for uncapacitated single allocation hub location in which pricing is taken into account. The objective is profit maximization by choosing the best hub and spoke topology and applying the optimal pricing, in the case of price-dependent demand. It is assumed that the source determining the number of hubs is endogenous. Two variants of the considered problem are addressed: deterministic and robust. For the initial non-linear model, we show how the deterministic variant can be reformulated as a mixed-integer linear program, excluding the price variables. In the robust counterpart case, the quantity of commodity flows between the pairs of customers is of stochastic nature. The goal of the robust variant is to design a hub and spoke network, together with the pricing structure, that would be immune to small perturbations of demand. Starting from the original model for the robust case, we have shown how to formulate an equivalent mixed-integer conic-quadratic program. In addition, we have proposed a 2-phase matheuristic approach for the robust variant. A computational study was conducted on the set of hub instances from the literature using the commercial state-of-the-art solver. The obtained computational results are thoroughly discussed, location patterns are analyzed and some managerial insights are provided. The experimental study also showed that the proposed matheuristic approach for the robust variant performs better compared to the commercial solver.

Data-Driven Robust Optimization for Steam Systems in Ethylene Plants under Uncertainty

In an ethylene plant, steam system provides shaft power to compressors and pumps and heats the process streams. Modeling and optimization of a steam system is a powerful tool to bring benefits and save energy for ethylene plants. However, the uncertainty of device efficiencies and the fluctuation of the process demands cause great difficulties to traditional mathematical programming methods, which could result in suboptimal or infeasible solution. The growing data-driven optimization approaches offer new techniques to eliminate uncertainty in the process system engineering community. A data-driven robust optimization (DDRO) methodology is proposed to deal with uncertainty in the optimization of steam system in an ethylene plant. A hybrid model of extraction–exhausting steam turbine is developed, and its coefficients are considered as uncertain parameters. A deterministic mixed integer linear programming model of the steam system is formulated based on the model of the components to minimize the operating cost of the ethylene plant. The uncertain parameter set of the proposed model is derived from the historical data, and the Dirichlet process mixture model is employed to capture the features for the construction of the uncertainty set. In combination with the derived uncertainty set, a data-driven conic quadratic mixed-integer programming model is reformulated for the optimization of the steam system under uncertainty. An actual case study is utilized to validate the performance of the proposed DDRO method.

New computational methods for classification problems in the existence of outliers based on conic quadratic optimization

Most of the statistical research involves classification which is a procedure utilized to establish prediction models to set apart and classify new observations in the dataset from every fields of science, technology, and economics. However, these models may give misclassification results when dataset contains outliers (extreme data points). Therefore, we dealt with outliers in classification problem: firstly, by combining robustness of mean-shift outlier model and then stability of Tikhonov regularization based on continuous optimization method called Conic Quadratic Programming. These new methodologies are performed on classification dataset within the existence of outliers, and the results are compared with parametric model by using well-known performance measures.

Fractional 0–1 programs: links between mixed-integer linear and conic quadratic formulations

This paper focuses on methods that improve the performance of solution approaches for multiple-ratio fractional 0–1 programs (FPs) in their general structure. In particular, we explore the links between equivalent mixed-integer linear programming and conic quadratic programming reformulations of FPs. Thereby, we show that integrating the ideas behind these two types of reformulations of FPs allows us to push further the limits of the current state-of-the-art results and tackle larger-size problems. We perform extensive computational experiments to compare the proposed approaches against the current reformulations from the literature.

A new approach to adaptive spline threshold autoregression by using Tikhonov regularization and continuous optimization

In this study adaptive spline threshold autoregression and conic quadratic programming is used to develope conic adaptive spline threshold autoregression. With the introduced approach the second stepwise algorithm of adaptive spline threshold autoregression model turned to the Tikhonov regularization problem which was transformed into conic quadratic programming problem. The aim is to attain an optimum solution chosen in many solutions obtained by determining the bounds of the optimization problem using multiobjective optimization approach. Furthermore, in application part we used two different data set to compare performances of linear regression, adaptive spline threshold autoregression and conic adaptive spline threshold autoregression approaches.