Approximation Algorithms for Scheduling on Multiple Machines

This paper deals with the capacitated single-stage production lot sizing and scheduling problem with multiple items, setup time, stochastic demand and unrelated parallel machines. A stochastic mixed-integer linear programming model is proposed to formulate the problem. Based on the uncertain constraints, the chance constrained programming approach is used to transform them into equivalent deterministic constraints and then obtain an optimal solution by deterministic mixed-integer linear programming model. Due to the complexity of problem, the mixed-integer programming (MIP) of the equivalence deterministic model of the capacitated single-stage production lot sizing problem with multiple items, setup time, stochastic demand and unrelated parallel machines is reformulated as a shortest path reformulation problem. The proposed algorithm is evaluated through a numerical example. Computational results show that the proposed method gives optimal or near optimal solution and have good-quality result for the test problem.

The next generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five control surfaces, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude of the aircraft. Next generation aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship no longer exists between the required control surface deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands to attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples the sensitivity of the solutions provided by the linear programming and quadratic programming methods.

One of the downsides of the massive multiple-input-multiple-output (M-MIMO) system is its computational complexity. Considering that techniques and different algorithms proposed in the literature applied to conventional MIMO may not be well suited or readily applicable to M-MIMO systems, in this paper, the application of different formulations inside the convex optimization framework is investigated. This paper is divided into two parts. In the first part, linear programming, quadratic programming (QP), and semidefinite programming are explored in an M-MIMO environment with high-order modulation and under realistic channel conditions, i.e., considering spatial correlation, error in the channel estimation, as well as different system loading. The bit error rate is evaluated numerically through Monte Carlo simulations. In the second part, algorithms to solve the QP formulation are explored, and computational complexity in terms of floating-point operations (flops) is compared with linear detectors. Those algorithms have interesting aspects when applied to our specific problem (M-MIMO detection formulated as QP), such as the exploitation of the structure of the problem ( simple constraints ) and the improvement of the rate of convergence due to the well-conditioned Gram matrix (channel hardening). The number of iterations is higher when the number of users $K$ becomes similar to the number of base station antennas $M$ (i.e., $K\approx M$ ) than the case $K\ll M$ ; the number of iterations increases slowly as the number of users $K$ and base station antennas $M$ increases while keeping a low system loading. The QP with projected algorithms presented better performance than minimum mean square error detector when $K\approx M$ and promising computational complexity for scenarios with increasing $K$ and low system loading.

Let there be a set M of m parallel machines and a set J of n jobs, where each job j takes \(p_{i,j}\) time units on machine \(M_i\). In makespan minimization the goal is to schedule each job non-preemptively on a machine such that the length of the schedule, the makespan, is minimum. We investigate a generalization of makespan minimization on unrelated parallel machines (\(R||C_{max}\)) where J is partitioned into b bags \(B=(B_1,\dots , B_b)\), and no two jobs belonging to the same bag can be scheduled on the same machine. First we present a simple b-approximation algorithm for \(R||C_{max}\) with bags (\(R|bag|C_{max}\)). Two machines \(M_i\) and \(M_{i'}\) have the same machine type if \(p_{i,j}=p_{i',j}\) for all \(j \in J\). We give a polynomial-time approximation scheme (PTAS) for \(R|bag|C_{max}\) with machine types where both the number of machine types and bags are constant. This result infers the existence of a PTAS for uniform parallel machines when the number of machine speeds and number of bags are both constant. Then, we present a b/2-approximation algorithm for the graph balancing problem with \(b \ge 2\) bags; the approximation ratio is tight for \(b=3\) unless Open image in new window and this algorithm solves the graph balancing problem with \(b=2\) bags in polynomial time. In addition, we present a polynomial-time algorithm for the restricted assignment problem on uniform parallel machines when all the jobs have unit length. To complement our algorithmic results, we show that when the jobs have lengths 1 or 2 it is Open image in new window -hard to approximate the makespan with approximation ratio less than 3/2 for both the restricted assignment and graph balancing problems with \(b=2\) bags and \(b=3\) bags, respectively. We also prove that makespan minimization on uniform parallel machines with \(b=2\) bags is strongly Open image in new window -hard.

We consider machine scheduling on unrelated parallel machines with the objective to minimize the schedule makespan. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a discrete renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of an integer linear programming formulation for a relaxation of the problem, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we show how to derive a 4-approximation algorithm. We also show how to tune our approach to yield a 3.75-approximation algorithm. This is achieved by applying the same rounding technique to a slightly modified linear programming relaxation, and by using a more sophisticated scheduling algorithm that is inspired by the harmonic algorithm for bin packing. We finally derive inapproximability results for two special cases, and discuss tightness of the integer linear programming relaxations.

The problem of regulating air traffic in the en route airspace of the National Airspace System is studied using a Eulerian network model to describe air traffic flow. The evolution of traffic on each edge of the network is modeled by a modified Lighthill-Whitham-Richards partial differential equation. The equation is transformed with a variable change, which makes it linear and enables us to use linear finite difference schemes to discretize the problem. We pose the problem of optimal traffic flow regulation as a continuous optimization program in which the partial differential equation appears in the constraints. We propose a discrete formulation of this problem, which makes all constraints (the discretized partial differential equations, boundary, and initial conditions) linear. Corresponding linear programming and quadratic programming based solutions to this convex optimization program yield globally optimal solutions to various air traffic management objectives. The proposed method is applied to the maximization of aircraft arrivals and minimization of delays in the arrival airspace due to exogenous capacity reductions. The corresponding linear and quadratic programs are solved numerically using CPLEX for a benchmark scenario in the Oakland Air Route Traffic Control Center. Several computational aspects of the method are assessed-in particular, accuracy of the numerical discretization, computational time, and storage space required by the method.

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidef-inite systems known to be strongly factorizable to a form of Cholesky-like factorization.Two different regularization techniques,primal; and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs.

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is, the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than 1.5 unless P = NP. We consider the case that there are only a constant number K of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for $$K=1$$ . We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any $$\varepsilon > 0$$ an assignment with makespan of length at most $$(1+\varepsilon )$$ times the optimum can be found in polynomial time in the input length and the exponent is independent of $$1/\varepsilon $$ . In particular, we achieve a running time of $$2^{{\mathcal {O}}(K\log (K) \nicefrac {1}{\varepsilon }\log ^4 \nicefrac {1}{\varepsilon })}+\mathrm {poly}(|I|)$$ , where |I| denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques.

It is sometimes necessary to consider more general nonlinear programs than the ones we have already studied: linear, quadratic, or conic programs. We give a brief introduction to this vast topic, and we discuss an application to estimating a volatility surface. Nonlinear Programming Consider a very general optimization problem of the form or the equivalent more concise form where. In the special case when all functions , hi are linear, problem (20.1) is a linear program as discussed in Chapter 2. When some of the functions , are nonlinear, problem (20.1) is a nonlinear program . Many practical problems are naturally formulated as nonlinear programs. We already saw quadratic programming and conic programming in earlier chapters. However, the family of problems that can be formulated as nonlinear programs is enormous, as many, if not most, imaginable kinds of constraints and objectives can be cast in terms of nonlinear functions. (For some noteworthy examples, see the exercises at the end of the chapter.) The immense modeling power of nonlinear programming comes at a cost: unlike linear, quadratic, and conic programming, which have a solid theory and are solvable via a few algorithmic templates, both the theory and methods to solve general nonlinear programs are far more complicated. Different types of nonlinear programs, determined by structural properties of the objective and constraint functions, are amenable to different types of algorithms. The subsequent sections sketch the main theory and most popular algorithmic ideas. A comprehensive treatment of this vast topic is beyond the scope of this textbook. We refer the reader to the excellent references Bertsekas (1999), Guler (2010), and Nocedal and Wright (2006) for more details. Numerical Nonlinear Programming Solvers There are numerous software packages for solving nonlinear programs. The following are some popular ones. We list them according to the class of algorithms (discussed in Section 20.4) that they are based on: (1) CONOPT, GRG2, Excel SOLVER. These solvers are based on the generalized reduced-gradient method . (2) MATLAB optimization toolbox, SNOPT, NLPQL. These solvers are based on sequential quadratic programming. (3) MINOS, LANCELOT. These solvers are based on an augmented Lagrangian approach. (4) MOSEK, LOQO, IPOPT. These solvers are based on interior-point methods.

Quadratic programming (QP) is a family of methods, techniques, and algorithms that can be used to minimize quadratic objective functions subject to linear constraints. QP shares many combinatorial features with linear programming (LP) and it is often used as the basis of constrained nonlinear programming. An important branch of QP is convex QP where the objective function is a convex quadratic function.

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

Frame permutation quantization (FPQ) is a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented and reconstruction algorithms based on linear programming and quadratic programming are derived. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions. Simulations for Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate.

Interregional and interseasonal competition in the U.S. beef industry have been studied extensively during the past decade. Linear programming, quadratic programming, simulation, and various other econometric models have formed the analytical frameworks in these studies.Reactive programming was first introduced as a useful tool in analyzing interregional competition problems by Tramel and Seale in 1959. Since that time, it has been utilized on a limited basis in spatial analyses in general, and the beef sector in particular. Goodwin used reactive programming in analyzing feeder cattle distribution patterns in the Southwest. In 1972, King and Ho reported a revised reactive programming algorithm and three illustrations of its applicability.

We investigate the NP-hard absolute value equation (AVE) Ax − |x|=b, where A is an arbitrary square matrix whose singular values exceed one. The significance of the absolute value equations arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear complementarity problem that in turn is equivalent to the absolute value equation. In this paper, after a lower uniform smooth approximation function of absolute value function is established, absolute value equation is transformed into smooth optimization problem, and solved by smoothing Newton method. The convergence of method is shown, and the preliminary numerical experiments are given.

Algorithms for hierarchical and semi-partitioned parallel scheduling