Optimality Tolerance Research Articles

Branch-and-price for routing with probabilistic customers

We study the Vehicle Routing Problem with Probabilistic Customers (VRP-PC), a two-stage optimization model, which is a fundamental building block within the broad family of stochastic routing problems. This problem is mainly motivated by logistics distribution networks in which customers receive frequent delivery services, and by the last mile problem faced by companies such as UPS and FedEx. In a first stage before customer service requests realize, a dispatcher determines a set of vehicle routes serving all potential customer locations. In a second stage occurring after observing all customer requests, vehicles execute planned routes skipping all locations of customers not requiring service. The objective is to minimize the expected vehicle travel cost assuming known customer realization probabilities. We propose a column generation framework to solve the VRP-PC to a given optimality tolerance. Specifically, we present two novel algorithms, one that under-approximates a solution’s expected cost, and another that uses its exact expected cost. Each algorithm is equipped with a route pricing mechanism that iteratively improves the approximation precision of a route’s reduced cost; this produces fast route insertions at the start of the algorithm and reaches termination conditions at the end of the execution. Compared to branch-and-cut algorithms for arc-based formulations, our framework can more readily incorporate sequence-dependent constraints, which are typically required in routing problems. We provide a priori and a posteriori performance guarantees for these algorithms, and demonstrate their effectiveness via a computational study on instances with realization probabilities ranging from 0.5 to 0.9.

A time-triggered dimension reduction algorithm for the task assignment problem

The task assignment problem is fundamental in combinatorial optimisation, aiming at allocating one or more tasks to a number of agents while minimizing the total cost or maximizing the overall assignment benefit. This problem is known to be computationally hard since it is usually formulated as a mixed-integer programming problem. In this paper, we consider a novel Time-Triggered Dimension Reduction Algorithm (TTDRA). We propose convexification approaches to convexify both the constraints and the cost function for the general non-convex assignment problem. The computational speed is accelerated via our time-triggered dimension reduction scheme, where the triggering condition is designed based on the optimality tolerance and the convexity of the cost function. Optimality and computational efficiency are verified via numerical simulations on benchmark examples.

Generalized stochastic Frank–Wolfe algorithm with stochastic “substitute” gradient for structured convex optimization

The stochastic Frank–Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the dependence on the optimality tolerance $$\varepsilon $$ in the guaranteed convergence rate for stochastic Frank–Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank–Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of “linear prediction” property (formally defined in the paper), which is typically present in loss minimization problems in practice. Our method also introduces the notion of a “substitute gradient” that is a not-necessarily-unbiased estimate of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank–Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank–Wolfe methods for a sufficiently small optimality tolerance, consistent with the theory developed herein.

Generalized Dual Dynamic Programming for Infinite Horizon Problems in Continuous State and Action Spaces

We describe a nonlinear generalization of dual dynamic programming (DP) theory and its application to value function estimation for deterministic control problems over continuous state and action spaces, in a discrete-time infinite horizon setting. We prove, using a Benders-type argument leveraging the monotonicity of the Bellman operator, that the result of a one-stage policy evaluation can be used to produce nonlinear lower bounds on the optimal value function that are valid over the entire state space. These bounds contain terms reflecting the functional form of the system's costs, dynamics, and constraints. We provide an iterative algorithm that produces successively better approximations of the optimal value function, and prove under certain assumptions that it achieves an arbitrarily low desired Bellman optimality tolerance at preselected points in the state space, in a finite number of iterations. We also describe means of certifying the quality of the approximate value function generated. We demonstrate the efficacy of the approach on systems whose dimensions are too large for conventional DP approaches to be practical.

A decoupled first/second-order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds

In order to be provably convergent towards a second-order stationary point, optimization methods applied to nonconvex problems must necessarily exploit both first and second-order information. However, as revealed by recent complexity analyses of some of these methods, the overall effort to reach second-order points is significantly larger when compared to the one of approaching first-order ones. On the other hand, there are other algorithmic schemes, initially designed with first-order convergence in mind, that do not appear to maintain the same first-order performance when modified to take second-order information into account. In this paper, we propose a technique that separately computes first and second-order steps, and that globally converges to second-order stationary points: it consists in better connecting the steps to be taken and the stationarity criteria, potentially guaranteeing larger steps and decreases in the objective. Our approach is shown to lead to an improvement of the corresponding complexity bound with respect to the first-order optimality tolerance, while having a positive impact on the practical behavior. Although the applicability of our ideas is wider, we focus the presentation on trust-region methods with and without derivatives, and motivate in both cases the interest of our strategy.

An experimental approach for examining solution errors of engineering problems with uncertain parameters

One potential overlook for applying optimization models to solve engineering problems is that their parameters are rarely error-free, implying that their solutions usually contain errors even when the models are solved to optimality. If the deviation between the solution based on parameters containing errors and the true optimal (but unavailable) solution based on error-free parameters is significant, the following decision-making could be meaningless. In this study, an experimental method is developed to evaluate solution errors of optimization models in which uncertain parameters are included in objective functions. A project scheduling problem is used as the case study. The effect of parameter errors and optimality tolerances in solution algorithms on solution errors are studied. The case study shows that the model solution errors increase as the scale of problem increases for the same range of parameter errors. It also shows that the model solution errors are similar for an optimality tolerance of within 4%. Regression models are estimated, which are useful for estimating potential errors between a solution based on parameters containing errors and the true optimal solution before a model is actually solved. They can also be used to determine values of optimality tolerance in solution algorithms that achieve the balance between solution quality and time.

Exact and Hybrid Solutions for the Multi-Objective VM Reassignment Problem

Machine Reassignment is a challenging problem for constraint programming (CP) and mixed integer linear programming (MILP) approaches, especially given the size of data centres. Hybrid solutions mixing CP and heuristic algorithms, such as, large neighbourhood search (CBLNS), also struggle to address the problem given its size and number of constraints. The multi-objective version of the Machine Reassignment Problem is even more challenging and it seems unlikely for CP, MILP or hybrid solutions to obtain good results in this context. As a result, the first approaches to address this problem have been based on other optimisation methods, including metaheuristics. In this paper we study three things: (i) under which conditions a mixed integer optimisation solver, such as IBM ILOG CPLEX, can be used for the Multi-objective Machine Reassignment Problem; (ii) how much of the search space can a well-known hybrid method such as CBLNS explore; and (iii) can we find a better hybrid approach combining MILP or CBLNS and an- other recent metaheuristic proposed for the problem (GeNePi). We show that MILP can handle only small or medium scale data centres, and with some relaxations, such as, an optimality tolerance gap and a limited number of directions explored in the search space. CBLNS on the other hand struggles with the problem in general but achieves reasonable performance for large instances of the problem. However, we show that our hybridisation improves both the quality of the set of solutions (CPLEX+GeNePi and CBLNS+GeNePi improve the solutions by +17.8% against CPLEX alone and +615% against CBLNS alone) and number of solutions (8.9 times more solutions than CPLEX alone and 56.76 times more solutions than CBLNS alone), while the processing time of CPLEX+GeNePi and CBLNS+GeNePi increases only by 6% and 16.4% respectively. Overall, the study shows that CPLEX+GeNePi is the best algorithm for small instances (CBLNS+GeNePi only gets 45.2% of CPLEX+GeNePi’s hypervolume) while CBLNS+GeNePi is better than the others on large instances (that CPLEX+GeNePi cannot address).

Optimizing environmental and economic impacts in supply chains in the FMCG industry

In this paper, the environmental impacts of a supply chain of a Fast Moving Consumer Goods (FMCG) company are considered. The environmental impacts are evaluated using the Eco-indicator 99. In the optimization of the tactical planning decisions both the environmental impacts and the total costs are considered using the ε-constraint method for identifying a set of Pareto-optimal solutions. For a case study containing 10 Stock-Keeping Units (SKUs), which was optimized with a 1% optimality tolerance, the environmental impacts could be reduced by 2.9% without increasing the total costs. A further reduction of environmental impacts of up to 6.3% was possible at an increase in total costs of 5.2%. An SKU decomposition algorithm was applied to optimize a larger case study containing 100 SKUs.

Maximizing strike aircraft planning efficiency for a given class of ground targets

Strike planning is one of the fundamental tasks of an advanced Air Force and involves the assignment of strike aircraft to ground targets with a maximum level of efficiency. Therefore, planning an optimal strike based on the preferences of the decision maker is crucial. The efficiency of the strike plan in this paper implies attacking the maximum number of targets while considering target priority and the desired level of damage on each target. The other objective is to minimize the cost of the strike plan. This paper develops a methodology that maximizes the efficiency of the strike plan. Given this efficiency, the aircraft and weapon costs plus the distance flown is then minimized. The methodology also considers the capacities for different types of aircraft and weapons at each aircraft base to avoid assigning aircraft to targets from a base where there are insufficient resources to do so. Computational results are presented that analyze the sensitivity of the model solution times to solver optimality tolerance and aircraft and weapon capacities. Results also suggest substantial cost savings are possible while still maintaining the effectiveness of the strike package.

Strategic robust supply chain design based on the Pareto-optimal tradeoff between efficiency and risk

The strategic design of a robust supply chain has to determine the configuration of the supply chain so that its performance remains of a consistently high quality for all possible future conditions. The current modeling techniques often only consider either the efficiency or the risk of the supply chain. Instead, we define the strategic robust supply chain design as the set of all Pareto-optimal configurations considering simultaneously the efficiency and the risk, where the risk is measured by the standard deviation of the efficiency. We model the problem as the Mean–Standard Deviation Robust Design Problem (MSD-RDP). Since the standard deviation has a square root expression, which makes standard maximization algorithms based on mixed-integer linear programming non-applicable, we show the equivalency to the Mean–Variance Robust Design Problem (MV-RDP). The MV-RDP yields an infinite number of mixed-integer programming problems with quadratic objective (MIQO) when considering all possible tradeoff weights. In order to identify all Pareto-optimal configurations efficiently, we extend the branch-and-reduce algorithm by applying optimality cuts and upper bounds to eliminate parts of the infeasible region and the non-Pareto-optimal region. We show that all Pareto-optimal configurations can be found within a prescribed optimality tolerance with a finite number of iterations of solving the MIQO. Numerical experience for a metallurgical case is reported.

On the Origin-Destination Demands Linear Programming Model for Network Revenue Management with Customer Choice

In this study, we research the problem of network revenue management with customer choice based on the Origin-Destination (O-D) demands. By dividing customers into different segments according to O-D pairs, we consider a network capacity control problem where each customer chooses the open product within the segment he belongs to. Starting with a Markov Decision Process (MDP) formulation, we approximate the value function with an affine function of the state vector. The affine function approximation results in a new Linear Program (LP) which yields tighter bounds than the Choice-based Deterministic Linear Program (CDLP). We give a column generation procedure for solving the LP within a desired optimality tolerance and present numerical results which show the policy perform from our solution approach can outperform that from the CDLP.

Sailing speed optimization for container ships in a liner shipping network

This paper first calibrates the bunker consumption – sailing speed relation for container ships using historical operating data from a global liner shipping company. It proceeds to investigate the optimal sailing speed of container ships on each leg of each ship route in a liner shipping network while considering transshipment and container routing. This problem is formulated as a mixed-integer nonlinear programming model. In view of the convexity, non-negativity, and univariate properties of the bunker consumption function, an efficient outer-approximation method is proposed to obtain an ε-optimal solution with a predetermined optimality tolerance level ε. The proposed model and algorithm is applied to a real case study for a global liner shipping company.

Optimal operating strategy for a long-haul liner service route

This paper proposes an optimal operating strategy problem arising in liner shipping industry that aims to determine service frequency, containership fleet deployment plan, and sailing speed for a long-haul liner service route. The problem is formulated as a mixed-integer nonlinear programming model that cannot be solved efficiently by the existing solution algorithms. In view of some unique characteristics of the liner shipping operations, this paper proposes an efficient and exact branch-and-bound based ε-optimal algorithm. In particular, a mixed-integer nonlinear model is first developed for a given service frequency and ship type; two linearization techniques are subsequently presented to approximate this model with a mixed-integer linear program; and the branch-and-bound approach controls the approximation error below a specified tolerance. This paper further demonstrates that the branch-and-bound based ε-optimal algorithm obtains a globally optimal solution with the predetermined relative optimality tolerance ε in a finite number of iterations. The case study based on an existing long-haul liner service route shows the effectiveness and efficiency of the proposed solution method.

An approximate dynamic programming approach for network capacity control problem with origin-destination demands

Bid-price control is a popular method for controlling the sale of inventory in revenue management. It is well known that the network capacity control problem can be formulated as a dynamic programming model. However, this formulation is intractable in practice due to the enormous size of the state space. As a result, various approximation methods are proposed in the literature. In this paper, we approximate the optimal dynamic programming value function with an affine function of the state vector and develop our model based on stochastic demand between origin-destination (O-D) pairs. We show that the resulting problem is the deterministic linear programming (DLP) for bid-price control. The DLP yields tighter bounds than the classical DLP. We give a column generation procedure for solving the DLP within a desired optimality tolerance, and present numerical results which show the policy perform from our solution approach can outperform that from the classical DLP.

Tactical and operational decisions for operating room planning: Efficiency and welfare implications

In this paper, we evaluate the impact on welfare implications of a 0-1 linear programming model to solve the Operating Room (OR) planning problem, taking a patient perspective. In particular, given a General Surgery Department made up of different surgical sub-specialties sharing a given number of OR block times, the model determines, during a given planning period, the allocation of those blocks to surgical sub-specialties, i.e. the so called Master Surgical Schedule Problem (MSSP), together with the subsets of elective patients to be operated on in each block time, i.e. the so called Surgical Case Assignment Problem (SCAP). The innovation of the model is two-fold. The first is that OR allocation is "optimal" if the available OR blocks are scheduled simultaneously to the proper subspecialty, at the proper time to the proper patient. The second is defining what "proper" means and include that in the objective function. In our approach what is important is not number of patients who can be treated in a given period but how much welfare loss, due to clinical deterioration or other negative consequences related to excessive waiting, can be prevented. In other words we assume a societal perspective in that we focus on "outcome" (health improving or preventing from worsening) rather than on "output" (delivered procedures). The model can be used both to develop weekly OR planning with given resources (operational decision), and to perform "what if" scenario analysis regarding how to increase the amount of OR time available for the entire department (tactical decision). The model performance is verified by applying it to a real scenario, the elective admissions of the General Surgery Department of the San Martino University Hospital in Genova (Italy). Despite the complexity of this NP-hard combinatorial optimization problem, computational results indicate that the model can solve all test problems within 600 s and an average optimality tolerance of less than 0.01%.

Relaxations of Weakly Coupled Stochastic Dynamic Programs

We consider a broad class of stochastic dynamic programming problems that are amenable to relaxation via decomposition. These problems comprise multiple subproblems that are independent of each other except for a collection of coupling constraints on the action space. We fit an additively separable value function approximation using two techniques, namely, Lagrangian relaxation and the linear programming (LP) approach to approximate dynamic programming. We prove various results comparing the relaxations to each other and to the optimal problem value. We also provide a column generation algorithm for solving the LP-based relaxation to any desired optimality tolerance, and we report on numerical experiments on bandit-like problems. Our results provide insight into the complexity versus quality trade-off when choosing which of these relaxations to implement.

Dynamic Bid Prices in Revenue Management

We formally derive the standard deterministic linear program (LP) for bid-price control by making an affine functional approximation to the optimal dynamic programming value function. This affine functional approximation gives rise to a new LP that yields tighter bounds than the standard LP. Whereas the standard LP computes static bid prices, our LP computes a time trajectory of bid prices. We show that there exist dynamic bid prices, optimal for the LP, that are individually monotone with respect to time. We provide a column generation procedure for solving the LP within a desired optimality tolerance, and present numerical results on computational and economic performance.

A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid

We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem’s class we study. We prove that under a certain mild assumption on the problem’s data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance $$\varepsilon$$grows as $$O(\sqrt{\ln \varepsilon^{-1}})$$.

A wireless ad hoc protocols comparison study: DSR, AODV and IWAR

Wireless ad hoc networks are relatively new and are gaining ground in research due to promises they offer. Wireless ad hoc networks do not require predefined configuration and have no fixed infrastructure. They are self-organizing and self-configuring networks. Several protocols have been developed that vary in the performance and complexity. Dynamic Source Routing (DSR) is a well-known pure on-demand protocol. It is a simple protocol and it was designed to have low overhead. But it does not select the optimal routes all the times. Ad Hoc On-Demand Distance-Vector (AODV) protocol is a wireless ad hoc protocol that makes its routing decision based on the distance vector. The Intelligent Wireless Ad hoc Routing (IWAR) is newly introduced protocol that it is also dynamic on-demand protocol. It implements more complex algorithm to select optimal routes and provide load balancing. This paper describes the DSR and AODV protocols, presents the design of the IWAR protocol, and compares the complexity, optimality, overhead traffic and fault tolerance of these protocols.

On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the function and subgradient values involves solving exactly a certain optimization problem, which can be relatively costly (and as we show, in any case unnecessary). We show that approximation with some finite precision is sufficient in this setting and give an explicit characterization of this precision. Alternatively, our result can be viewed as a stability analysis of standard proximal bundle methods, as it answers the following question: for a given approximation error, what kind of approximate solution can be obtained and how does it depend on the magnitude of the perturbation?