A stopping rule while searching for optimal solution of facility-location

Solutions to combinatorial optimization, such as p-median problems of locating facilities, frequently rely on heuristics to minimize the objective function. The minimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. However, pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small branch of the literature suggests using statistical principles to estimate the minimum and use the estimate for either stopping or evaluating the quality of the solution. In this paper we use test-problems taken from Baesley's OR-library and apply Simulated Annealing on these p-median problems. We do this for the purpose of comparing suggested methods of minimum estimation and, eventually, provide a recommendation for practioners. An illustration ends the paper being a problem of locating some 70 distribution centers of the Swedish Post in a region.

Localization of the optimal solution and a posteriori bounds for aggregation

After an aggregated problem has been solved, it is often desirable to estimate the accuracy loss due to the fact that a simpler problem than the original one has been solved. One way of measuring this loss in accuracy is the difference in objective function values. To get the bounds for this difference, Zipkin (Operations Research 1980;28:406) has assumed, that a simple (knapsack-type) localization of an original optimal solution is known. Since then various extensions of Zipkin's bound have been proposed, but under the same assumption. A method to compute the bounds for variable aggregation for convex problems, based on general localization of the original solution is proposed. For some classes of the original problem it is shown how to construct the localization. Examples are given to illustrate the main constructions and a small numerical study is presented.

Numerical solution of optimal control problems with constant control delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a stateof- the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N0), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H2, is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N0, M, and n. Asymptotic approximations of H2 are deduced and tested for both one-dimensional (1-D) and 2-D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power-law relationship, C= aMNb0, with the CPU cost index, b, indicating the weighting of N0 in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N0 determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015

Optimal power system stabilizer tuning in multi-machine system via an improved differential evolution

Power system stabilizer (PSS) is one of the most important controllers in modern power systems for damping low frequency oscillations. Many efforts have been dedicated to design the tuning methodologies and allocation techniques to obtain optimal damping behaviors of the system. Traditionally, it is tuned mostly for local damping performance, however, in order to obtain a globally optimal performance, the tuning of PSS needs to be done considering more variables. Furthermore, with the enhancement of system interconnection and the increase of system complexity, new tools are required to achieve global tuning and coordination of PSS to achieve optimal solution in a global meaning. Differential evolution (DE) is a recognized as a simple and powerful global optimum technique, which can gain fast convergence speed as well as high computational efficiency. However, as many other evolutionary algorithms (EA), the premature of population restricts optimization capacity of DE. In this paper, a modified DE is proposed and applied for optimal PSS tuning of 39-Bus New-England system. New operators are introduced to reduce the probability of getting premature. To investigate the impact of system conditions on PSS tuning, multiple operating points will be studied. Simulation result is compared with standard DE and particle swarm optimization (PSO).

Output feedback optimal control with contraints

In Chapters 1 through 9 of the book (with the exception of a brief discussion on observers and integral action in Section 5.5 of Chapter 5) we considered constrained optimal control problems for systems without uncertainty, that is, with no unmodelled dynamics or disturbances, and where the full state was available for measurement. More realistically, however, it is necessary to consider control problems for systems with uncertainty. This chapter addresses some of the issues that arise in this situation. As in Chapter 9, we adopt a stochastic description of uncertainty, which associates probability distributions to the uncertain elements, that is, disturbances and initial conditions. (See Section 12.6 for references to alternative approaches to model uncertainty.) When incomplete state information exists, a popular observer-based control strategy in the presence of stochastic disturbances is to use the certainty equivalence [CE] principle, introduced in Section 5.5 of Chapter 5 for deterministic systems. In the stochastic framework, CE consists of estimating the state and then using these estimates as if they were the true state in the control law that results if the problem were formulated as a deterministic problem (that is, without uncertainty). This strategy is motivated by the unconstrained problem with a quadratic objective function, for which CE is indeed the optimal solution (˚Astr¨om 1970, Bertsekas 1976). One of the aims of this chapter is to explore the issues that arise from the use of CE in RHC in the presence of constraints. We then turn to the obvious question about the optimality of the CE principle. We show that CE is, indeed, not optimal in general. We also analyse the possibility of obtaining truly optimal solutions for single input linear systems with input constraints and uncertainty related to output feedback and stochastic disturbances.We first find the optimal solution for the case of horizon N = 1, and then we indicate the complications that arise in the case of horizon N = 2. Our conclusion is that, for the case of linear constrained systems, the extra effort involved in the optimal feedback policy is probably not justified in practice. Indeed, we show by example that CE can give near optimal performance. We thus advocate this approach in real applications.

Optimal eradication: When to stop looking for an invasive plant

The notion of being sure that you have completely eradicated an invasive species is fanciful because of imperfect detection and persistent seed banks. Eradication is commonly declared either on an ad hoc basis, on notions of seed bank longevity, or on setting arbitrary thresholds of 1% or 5% confidence that the species is not present. Rather than declaring eradication at some arbitrary level of confidence, we take an economic approach in which we stop looking when the expected costs outweigh the expected benefits. We develop theory that determines the number of years of absent surveys required to minimize the net expected cost. Given detection of a species is imperfect, the optimal stopping time is a trade-off between the cost of continued surveying and the cost of escape and damage if eradication is declared too soon. A simple rule of thumb compares well to the exact optimal solution using stochastic dynamic programming. Application of the approach to the eradication programme of Helenium amarum reveals that the actual stopping time was a precautionary one given the ranges for each parameter. © 2006 Blackwell Publishing Ltd/CNRS.

Optimal eradication: when to stop looking for an invasive plant

The notion of being sure that you have completely eradicated an invasive species is fanciful because of imperfect detection and persistent seed banks. Eradication is commonly declared either on an ad hoc basis, on notions of seed bank longevity, or on setting arbitrary thresholds of 1% or 5% confidence that the species is not present. Rather than declaring eradication at some arbitrary level of confidence, we take an economic approach in which we stop looking when the expected costs outweigh the expected benefits. We develop theory that determines the number of years of absent surveys required to minimize the net expected cost. Given detection of a species is imperfect, the optimal stopping time is a trade-off between the cost of continued surveying and the cost of escape and damage if eradication is declared too soon. A simple rule of thumb compares well to the exact optimal solution using stochastic dynamic programming. Application of the approach to the eradication programme of Helenium amarum reveals that the actual stopping time was a precautionary one given the ranges for each parameter.

A Modified Dubins Method for Optimal Path Planning of a Miniature Air Vehicle Converging to a Straight Line Path

This paper presents a Dubins model based strategy to determine the optimal path of a Miniature Air Vehicle (MAV), constrained by a bounded turning rate, that would enable it to fly along a given straight line, starting from an arbitrary initial position and orientation. The method is then extended to meet the same objective in the presence of wind which has a magnitude comparable to the speed of the MAV. We use a modification of the Dubins' path method to obtain the complete optimal solution to this problem in all its generality.

A generalized Linear Programming based approach to optimal divisible load scheduling

In this paper we propose a general Linear Programming (LP) based formulation and solution methodology for obtaining optimal solution to the load distribution problem in divisible load scheduling. We exploit the power of the versatile LP formulation to propose algorithms that yield exact solutions to several very general load distribution problems for which either no solutions or only heuristic solutions were available. We consider both star (single-level tree) networks and linear daisy chain networks, having processors equipped with front-ends, that form the generic models for several important network topologies. We consider arbitrary processing node availability or release times and general models for communication delays and computation time that account for constant overheads such as start up times in communication and computation. The optimality of the LP based algorithms is proved rigorously.

Tunable Locally-Optimal Geographical Forwarding in Wireless Sensor Networks with Sleep-Wake Cycling Nodes

We consider a wireless sensor network whose main function is to detect certain infrequent alarm events, and to forward alarm packets to a base station, using geographical forwarding. The nodes know their locations, and they sleep-wake cycle, waking up periodically but not synchronously. In this situation, when a node has a packet to forward to the sink, there is a trade-off between how long this node waits for a suitable neighbor to wake up and the progress the packet makes towards the sink once it is forwarded to this neighbor. Hence, in choosing a relay node, we consider the problem of minimizing average delay subject to a constraint on the average progress. By constraint relaxation, we formulate this next hop relay selection problem as a Markov decision process (MDP). The exact optimal solution (BF (Best Forward)) can be found, but is computationally intensive. Next, we consider a mathematically simplified model for which the optimal policy (SF (Simplified Forward)) turns out to be a simple one-step-look-ahead rule. Simulations show that SF is very close in performance to BF, even for reasonably small node density. We then study the end-to-end performance of SF in comparison with two extremal policies: Max Forward (MF) and First Forward (FF), and an end-to-end delay minimising policy proposed by Kim et al. 1]. We find that, with appropriate choice of one hop average progress constraint, SF can be tuned to provide a favorable trade-off between end-to-end packet delay and the number of hops in the forwarding path.

Optimal association of mobile wireless devices with a WLAN-3G access network

In this paper, we consider the problem of association of wireless stations (STAs) with an access network served by a wireless local area network (WLAN) and a 3G cellular network. There is a set of WLAN Access Points (APs) and a set of 3G Base Stations (BSs) and a number of STAs each of which needs to be associated with one of the APs or one of the BSs. We concentrate on downlink bulk elastic transfers. Each association provides each ST with a certain transfer rate. We evaluate an association on the basis of the sum log utility of the transfer rates and seek the utility maximizing association. We also obtain the optimal time scheduling of service from a 3G BS to the associated STAs. We propose a fast iterative heuristic algorithm to compute an association. Numerical results show that our algorithm converges in a few steps yielding an association that is within 1% (in objective value) of the optimal (obtained through exhaustive search); in most cases the algorithm yields an optimal solution.

Optimal trajectory planning for path convergence in three-dimensional space

This article addresses the problem of determining the shortest path that connects a given initial configuration (position, heading angle, and flight path angle) to a given rectilinear or a circular path in three-dimensional space for a constant speed and turn-rate constrained aerial vehicle. The final path is assumed to be located relatively far from the starting point. Due to its simplicity and low computational requirements the algorithm can be implemented on a fixed-wing type unmanned air vehicle in real time in missions where the final path may change dynamically. As wind has a very significant effect on the flight of small aerial vehicles, the method of optimal path planning is extended to meet the same objective in the presence of wind comparable to the speed of the aerial vehicles. But, if the path to be followed is closer to the initial point, an off-line method based on multiple shooting, in combination with a direct transcription technique, is used to obtain the optimal solution. Optimal paths are generated for a variety of cases to show the efficiency of the algorithm. Simulations are presented to demonstrate tracking results using a 6-degrees-of-freedom model of an unmanned air vehicle.

BELIEF PROPAGATION FOR OPTIMAL EDGE COVER IN THE RANDOM COMPLETE GRAPH

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and Wastlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the (2) limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.

Pulse vs. Optimal Stationary Fishing: The Northern Stock of Hake

Pulse fishing may be a global optimal strategy in multicohort fisheries. In this article we compare the pulse fishing solutions obtained by using global numerical methods with the analytical stationary optimal solution. This allows us to quantify the potential benefits associated with the use of periodic fishing in the Northern Stock of hake. Results show that: first, management plans based exclusively on traditional reference targets as Fmsy may drive fishery economic results far from the optimal; second, global optimal solutions would imply, in a cyclical manner, the closure of the fishery for some periods and third, second best stationary policies with stable employment only reduce optimal present value of discounted profit in a 2%.