Sequence Of Subproblems Research Articles

A Computation-Efficient Approach for Segment Routing Traffic Engineering

The unprecedented growth of network traffic has brought excessive challenges to network operators. To prevent network congestion, network operators conduct traffic engineering (TE) for their routing optimization. In recent years, segment routing traffic engineering (SRTE) has emerged as one of the promising approaches for its high scalability and low control overheads. However, conventional SRTE approaches in large-scale networks are computationally prohibitive, which may lead to delayed system operations and unsatisfactory service qualities. In this paper, we formulate a bi-objective mixed-integer nonlinear program (BOMINLP) to investigate the trade-off between link utilization and computation time in SRTE. Due to the difficulty in solving the original problem directly, we decompose it into two sequential sub-problems. The first sub-problem is to minimize computation time through node selection, and the second one is to minimize maximum link utilization via flow assignment. To this end, we first employ randomized sampling based on stretch bounding to obtain a reduced solution space and then solve a linear program (LP) using existing software tools for the sub-problems. To evaluate our proposed solution, we employ network topologies and traffic matrices from publicly available datasets. Our simulation results show that our proposed solution can effectively reduce computation time while retaining comparable maximum link utilization as compared with several comparison approaches.

An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints

Vehicle routing problem (VRP) is a typical and important combinatorial optimization problem, and is often involved with complicated temporal and spatial constraints in practice. In this paper, the VRP is formulated as an optimization model for minimizing the number of vehicles and the total transportation cost subject to constraints on loading plan, service time and weight capacity. The transportation cost consists of the rent charge of vehicles, fuel cost, and carbon tax. Owing to complexity of the built model, it is divided into two subproblems by a two-stage optimization approach: at the first stage, the number of vehicles is minimized, then the routing plan is optimized at the second stage. For solving the sequential subproblems, two correlated genetic algorithms are developed, which share the same initial population to reduce their computational costs. Numerical results indicate that the developed algorithms are efficient, and a number of important managerial insights are revealed from the model.

Optimal Design of Resource Element Mapping for Sparse Spreading Non-Orthogonal Multiple Access

We propose a new design of the optimal resource element (RE) mapping patterns to maximize the sum-rate of a sparse spreading non-orthogonal multiple access system. In this system, the grouped users who use the same radio resource have different channel conditions. First, we formulate a sum-rate optimization problem subject to sparsity and power constraints. To solve this non-trivial optimization problem, we transform it to an equivalent penalized problem by deriving the closed-form penalty parameters. We then convert this problem to a sequence of subproblems using the difference of convex programming and propose an efficient algorithm to find the optimal solution. Numerical results demonstrate that the proposed algorithm achieves a higher sum-rate and a superior block error rate performance than the conventional RE mapping schemes.

A Sequential Partial Optimization Algorithm with Guaranteed Convergence for Minimax Design of IIR Digital Filters

Challenges for optimal design of infinite impulse response digital filters include the high nonconvexity of design problem and inevitable stability constraints on the filters. To reduce the nonconvexity and tackle the stability constraints, a sequential partial optimization (SPO) algorithm was recently developed to divide the design problem into a sequence of subproblems, each updating only two second-order denominator factors. But the convergence of that algorithm is not guaranteed. By applying an incremental update with an optimized step length in each subproblem, this paper presents an improved SPO algorithm which is guaranteed to converge to a Karush–Kuhn–Tucker (not necessarily global) solution of the design problem. This paper also extends the SPO algorithm to a more general case where the number of denominator factors optimized in the subproblems can be any positive number smaller than half of the denominator order. Convergence performance of the algorithm is shown by the design of two example filters with typical specifications widely adopted in the literature. Comparisons with state-of-the-art methods demonstrate that the improved SPO algorithm obtains better filters than the competing methods in terms of the maximum magnitude of frequency-response error.

The Convex Feasible Set Algorithm for Real Time Optimization in Motion Planning

With the development of robotics, there are growing needs for real time motion planning. However, due to obstacles in the environment, the planning problem is highly nonconvex, which makes it difficult to achieve real time computation using existing nonconvex optimization algorithms. This paper introduces the convex feasible set algorithm, which is a fast algorithm for nonconvex optimization problems that have convex costs and nonconvex constraints. The idea is to find a convex feasible set for the original problem and iteratively solve a sequence of subproblems using the convex constraints. The feasibility and the convergence of the proposed algorithm are proved in the paper. The application of this method on motion planning for mobile robots is discussed. The simulations demonstrate the effectiveness of the proposed algorithm.

Stable and efficient resource allocation under weak priorities

We study the indivisible object allocation problem without monetary transfer, in which each object is endowed with a weak priority ordering over agents. It is well known that stability is generally not compatible with efficiency. We characterize the priority structures for which a stable and efficient assignment always exists, as well as the priority structures that admit a stable, efficient and (group) strategy-proof rule. While house allocation problems and housing markets are two classic families of allocation problems that admit a stable, efficient and group strategy-proof rule, any priority-augmented allocation problem with more than three objects admits such a rule if and only if it is decomposable into a sequence of subproblems, each of which has the structure of a house allocation problem or a housing market.

An efficient heuristic algorithm for the alternative-fuel station location problem

We have developed an efficient heuristic algorithm for location of alternative-fuel stations. The algorithm is constructed based on solving the sequence of subproblems restricted on a set of promising station candidates, and fixing a number of the best promising station locations. The set of candidates is initially determined by solving a relaxation model, and then modified by exchanging some stations between the promising candidate set and the remaining station set. A number of the best station candidates in the promising candidate set can be fixed to improve computation time. In addition, a parallel computing strategy is integrated into solving simultaneously the set of subproblems to speed up computation time. Experimental results carried out on the benchmark instances show that our algorithm outperforms genetic algorithm and greedy algorithm. As compared with CPLEX solver, our algorithm can obtain all the optimal solutions on the tested instances with less computation time.

Allocating products on shelves under merchandising rules: Multi-level product families with display directions

Retailers’ individual products are categorized as part of product families. Merchandising rules specify how the products should be arranged on the shelves using product families, creating more structured displays capable of increasing the viewers’ attention. This paper presents a novel mixed integer programming formulation for the Shelf Space Allocation Problem considering two innovative features emerging from merchandising rules: hierarchical product families and display directions. The formulation uses single commodity flow constraints to model product sequencing and explores the product families’ hierarchy to reduce the combinatorial nature of the problem. Based on the formulation, a mathematical programming-based heuristic was also developed that uses product families to decompose the problem into a sequence of sub-problems. To improve performance, its original design was adapted following two directions: recovery from infeasible solutions and reduction of solution times. A new set of real case benchmark instances is also provided, which was used to assess the formulation and the matheuristic. This approach will allow retailers to efficiently create planograms capable of following merchandising rules and optimizing shelf space revenue.

Comonotonic approximation to periodic investment problems under stochastic drift

We investigate periodic investment problems under a Black–Scholes market with stochastic drift. The decision maker invests a series of positive amounts at finitely predetermined time spots, to maximize the expected terminal wealth while controlling its downside risk as measured by the Condition Value at Risk (CVaR). It turns out that the increment for unit wealth on the whole path can be divided into two parts: the increment corresponding to the stochastic drift and that corresponding to the Brownian Motion. A comonotonic approximation is proposed for the second part, and an upper bound is provided for the CVaR of the first part, which construct together a closed-form approximation of the terminal wealth under the risk measure of CVaR. We further decompose the problem into a sequence of sub-problems whose optimal solutions are explicit and follow fractional Kelly Strategy. Numerical and empirical results illustrate the performance of our methodology.

Reliability-based topology optimization of uncertain building systems subject to stochastic excitation

Topology optimization has traditionally been developed in a deterministic setting, notwithstanding the considerable uncertainties that generally affect both the system as well as the excitation. Therefore, the development of methods that are capable of describing the performance of uncertain systems in a fully probabilistic setting would represent an important step forward. In particular, the ability to consider reliability constraints written in terms of first excursion probabilities posed on systems driven by general stochastic excitation would allow a wide variety of important design scenarios to be modeled. This paper is focused on proposing a simulation-centered reliability-based topology optimization framework to this end. In particular, an approach is developed based on defining, from the argumentation of the simulation process, an optimization sub-problem that not only approximately decouples the probabilistic analysis from the optimization loop, but takes a form that can be extremely efficiently solved. By solving a limited sequence of sub-problems, solutions are found that rigorously meet the first excursion constraints of the original problem. A series of case studies are presented illustrating the potential of the proposed framework.

Cyclic best first search: Using contours to guide branch‐and‐bound algorithms

AbstractThe cyclic best‐first search (CBFS) strategy is a recent search strategy that has been successfully applied to branch‐and‐bound algorithms in a number of different settings. CBFS is a modification of best‐first search (BFS) that places search tree subproblems into contours which are collections of subproblems grouped in some way, and repeatedly cycles through all non‐empty contours, selecting one subproblem to explore from each. In this article, the theoretical properties of CBFS are analyzed for the first time. CBFS is proved to be a generalization of all other search strategies by using a contour definition that explores the same sequence of subproblems as any other search strategy. Further, a bound is proved between the number of subproblems explored by BFS and the number of children generated by CBFS, given a fixed branching strategy and set of pruning rules. Finally, a discussion of heuristic contour‐labeling functions is provided, and proof‐of‐concept computational results for mixed‐integer programming problems from the MIPLIB 2010 database are shown. © 2017 Wiley Periodicals, Inc. Naval Research Logistics, 64: 64–82, 2017

Learning decision rules using adaptive technologies: a hybrid approach based on sequential covering

Abstract: Sequential covering strategies are commonly used in rule learning algorithms, as they apply separate and conquer approaches, in which the task of finding a complete rule base is reduced to a sequence of subproblems; each solution to a subproblem consists in adding a single rule. We propose an alternative for rule learning, namely the use of adaptive devices whose behavior is defined by a dynamic set of rules. In order to integrate features from rule learning methods and adaptive technologies, this paper presents a hybrid approach to learn classification rules, such that the set of rules is dynamically modified by adding or removing rules. The results have been promising towards applying adaptive technology to learn rules directly from data.

A Sequential Two-Step Algorithm for Fast Generation of Vehicle Racing Trajectories

The problem of maneuvering a vehicle through a race course in minimum time requires computation of both longitudinal (brake and throttle) and lateral (steering wheel) control inputs. Unfortunately, solving the resulting nonlinear optimal control problem is typically computationally expensive and infeasible for real-time trajectory planning. This paper presents an iterative algorithm that divides the path generation task into two sequential subproblems that are significantly easier to solve. Given an initial path through the race track, the algorithm runs a forward–backward integration scheme to determine the minimum-time longitudinal speed profile, subject to tire friction constraints. With this fixed speed profile, the algorithm updates the vehicle's path by solving a convex optimization problem that minimizes the resulting path curvature while staying within track boundaries and obeying affine, time-varying vehicle dynamics constraints. This two-step process is repeated iteratively until the predicted lap time no longer improves. While providing no guarantees of convergence or a globally optimal solution, the approach performs very well when validated on the Thunderhill Raceway course in Willows, CA. The predicted lap time converges after four to five iterations, with each iteration over the full 4.5 km race course requiring only 30 s of computation time on a laptop computer. The resulting trajectory is experimentally driven at the race circuit with an autonomous Audi TTS test vehicle, and the resulting lap time and racing line are comparable to both a nonlinear gradient descent solution and a trajectory recorded from a professional racecar driver. The experimental results indicate that the proposed method is a viable option for online trajectory planning in the near future.

A coordinate descent homotopy method for linearly constrained nonsmooth convex minimization

A problem in optimization, with a wide range of applications, entails finding a solution of a linear equation with various minimization properties. Such applications include compressed sensing, which requires an efficient method to find a minimal norm solution. We propose a coordinate descent homotopy method to solve the linearly constrained convex minimization problem where P is proper, convex and lower semicontinuous. A well-known special case is the basis pursuit problem . The greedy-type coordinate descent method is applied to solve the regularized linear least squares problem, which arises as a sequence of subproblems for the proposed method, and we show global linear convergence. We report numerical results for solving large-scale basis pursuit problem. Comparison with Bregman iterative algorithm [W. Yin, S. Osher, D. Goldfarb, and J. Darbon, Bregman iterative algorithms for -minimization with applications to compressed sensing, SIAM J. Image Sci. 1 (2008), pp. 143–168] and linearized Bregman iterative algorithm [J.-F. Cai, S. Osher, and Z. Shen, Linearized Bregman iterations for compressed sensing, Math. Comput. 78 (2009), pp. 1515–1536] suggests that the proposed method can be used as an efficient method for minimization problem.

Application of Sustainable Integrated Process Design and Control for Four Distillation Column Systems

The objective of this paper is to present the application of sustainable integrated process design and control methodology for distillation columns systems. The sustainable integrated process design and control methodology is developed and able to find the optimal solution for a single distillation column to ensure the design is more cost efficient, controllable and sustainable to meet the product quality. The sustainable integrated process design and control problem for a distillation column system is typically formulated as a mathematical programming (optimization with constraints) problem, and solved by decomposing it to six sequential hierarchical sub-problems: (i) pre-analysis, (ii) design analysis, (iii) controller design analysis, (iv) sustainability analysis, (v) detailed economics analysis and (vi) final selection and verification. In the pre-analysis sub-problem, the concept of driving force is used to locate the optimal design-control-sustainable solution targets, which are defined at the maximum point of the driving force diagram. The sustainability analysis sub-problem was analysed by using the three-dimensional sustainability index. The results through separation of n-Pentane, n-Hexane, n-Heptane, n-Octane and n-Decane mixtures with four distillation column systems shows the methodology is capable to find the optimal solution for multiple distillation column systems that satisfies design, control and sustainability criteria in a simple and efficient way.

Genetic single objective optimisation for sizing and allocation of renewable DG systems

The optimal design of renewable-based distributed generations (DGs) is a challenging issue in order to maximise their benefits and to overcome power quality problems. Therefore, this paper proposes a methodology for optimal allocation and sizing of renewable DG units to minimise total power losses over radial distribution systems. The planning problem is formulated as a single objective nonlinear mixed integer-constrained optimisation problem and is solved by using the augmented Lagrangian genetic algorithm (ALGA) by combining the objective function and the nonlinear constraints. In that case, the ALGA solves a sequence of sub-problems where the objective function penalises the constraints violation in order to obtain the best solution. The proposed technique is applied to IEEE radial test systems including 33-bus, 69-bus and 119-bus and is implemented with different scenarios including all possible combinations and various types of renewable DG units to prove the effectiveness of the proposed methodology.

A decompose-and-fix heuristic based on multi-commodity flow models for driver rostering with days-off pattern

Facing severe budgetary constraints, public transport companies are forced to efficiently manage staff, one of the most expensive resources.The driver rostering problem in a public transit company consists of defining rosters, that is, assigning daily crew duties to the company's drivers for a pre-defined time horizon, ensuring transport demand in a specific area, at low operating costs, while complying with legal restrictions and agreements between the company and the driver unions.The objective of this paper is the study of mathematical models and optimization techniques that lead to new computational tools able to solve the bus driver rostering problem with days-off patterns by producing solutions that increase efficiency and reduce operating costs, while improving or maintaining the service quality and balancing the drivers’ workload.Three mixed integer linear programming formulations are presented and compared from a theoretical point of view: an assignment/cover model, a multi-commodity flow model and a new multi-commodity flow/assignment model.Based on a hierarchy of the decisions made during the resolution of the problem, a new decompose-and-fix heuristic is developed by exploring the underlying structure of the multi-commodity flow models. The heuristic solves the sequence of sub-problems identified by the hierarchy, while fixing or bounding the value of some variables to incorporate previous decisions.Computational experiments were carried out with instances, derived from real world data and from benchmark data, characterized by a days-off pattern in use at two Portuguese public bus transport companies. Computational results confirm the good performance of the decompose-and-fix heuristic.

Throughput Maximization with S-graph Framework using Global Branching Tree

Abstract Majozi and Friedler [3] developed the first approach for throughput maximization, which has no need for the optimal number of time points a-priori. The feasibility of a sequence of subproblems is examined with guided search driven S-graph based approach. Although, the approach has proved to be efficient for several problems [1], it can be developed further to increase the size of addressable cases. In the original approach, a branching tree is generated for each subproblem separately, even though these trees can have significantly large overlaps. The presented approach exploits this redundancy to reduce the computational needs for industrial scale problems.

Convex mixed integer nonlinear programming problems and an outer approximation algorithm

In this paper, we mainly study one class of convex mixed-integer nonlinear programming problems (MINLPs) with non-differentiable data. By dropping the differentiability assumption, we substitute gradients with subgradients obtained from KKT conditions, and use the outer approximation method to reformulate convex MINLP as one equivalent MILP master program. By solving a finite sequence of subproblems and relaxed MILP problems, we establish an outer approximation algorithm to find the optimal solution of this convex MINLP. The convergence of this algorithm is also presented. The work of this paper generalizes and extends the outer approximation method in the sense of dealing with convex MINLPs from differentiable case to non-differentiable one.

Joint Information- and Jamming-Beamforming for Physical Layer Security With Full Duplex Base Station

In this paper, we design joint information beamforming and jamming beamforming to guarantee both transmit security and receive security for a full-duplex base-station (FD-BS). Specifically, we aim to maximize the secret transmit rate while constrain the secret receive rate to be greater than a predefined bound. When FD-BS is equipped with a single transmit antenna, we derive the optimal solutions in closed-form, and interestingly, the results say that simultaneous information and jamming transmission cannot be optimal. When FD-BS is equipped with multiple transmit antennas, we convert the original nonconvex problem into a sequence of subproblems where the semidefinite programming (SDP) relaxation can be applied to efficiently find the optimal solutions. We strictly prove that such a relaxation does not change the optimality for these subproblems. Then the global optimal solutions of the original nonconvex problem can be obtained via a one-dimensional search. Simulation results are provided to verify the efficiency of the proposed algorithms.