Near Optimality of Linear Delayed Doubly Stochastic Control Problem

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.

Studying the effect of Eliminating Repeated Individuals from the Population in a Genetic Algorithm: Solution Perspectives for the Travelling Salesman Problem

The Travelling Salesman Problem (TSP) is one of the main routing problems in the Logistics and Supply Chain Management fields. Given its computational complexity, metaheuristics are frequently needed to solve it to near-optimality. In this aspect, Genetic Algorithms (GA) are promising methods, however, their search performance depends of populations of solutions which can increase computational processing. Thus, the management of this component is subject to adaptations to reduce its computational burden and improve overall performance. This work explores on the elimination of repeated individuals within the population which may represent a significant fraction of its size and do not add valuable information to the solution search mechanisms of the GA. This cleaning process is expected to contribute to solution diversity. Experiments performed with different TSP test instances support the finding that this cleaning process can improve the convergence of the GA to very suitable solutions (within the 10% error limit). These findings were statistically validated.

Local Turnpike Analysis Using Local Dissipativity for Discrete Time Discounted Optimal Control

Recent results in the literature have provided connections between the so-called turnpike property, near optimality of closed-loop solutions, and strict dissipativity. Motivated by applications in economics, optimal control problems with discounted stage cost are of great interest. In contrast to non-discounted optimal control problems, it is more likely that several asymptotically stable optimal equilibria coexist. Due to the discounting and transition cost from a local to the global equilibrium, it may be more favourable staying in a local equilibrium than moving to the global—cheaper—equilibrium. In the literature, strict dissipativity was shown to provide criteria for global asymptotic stability of optimal equilibria and turnpike behaviour. In this paper, we propose a local notion of discounted strict dissipativity and a local turnpike property, both depending on the discount factor. Using these concepts, we investigate the local behaviour of (near-)optimal trajectories and develop conditions on the discount factor to ensure convergence to a local asymptotically stable optimal equilibrium.

Complexity of near-optimal robust versions of multilevel optimization problems

Near-optimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of near-optimal robust multilevel problems, where near-optimal robustness is modelled through additional adversarial decision-makers. Near-optimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without near-optimality robustness under general conditions.

Dissecting the duality gap: the supporting hyperplane interpretation revisited

We revisit the classic supporting hyperplane illustration of the duality gap for non-convex optimization problems. It is refined by dissecting the duality gap into two terms: the first measures the degree of near-optimality in a Lagrangian relaxation, while the second measures the degree of near-complementarity in the Lagrangian relaxed constraints. We also give an example of how this dissection may be exploited in the design of a solution approach within discrete optimization.

Practical Efficient Regional Land-Use Planning Using Constrained Multi-Objective Genetic Algorithm Optimization

Practical efficient regional land-use planning requires planners to balance competing uses, regional policies, spatial compatibilities, and priorities across the social, economic, and ecological domains. Genetic algorithm optimization has progressed complex planning, but challenges remain in developing practical alternatives to random initialization, genetic mutations, and to pragmatically balance competing objectives. To meet these practical needs, we developed a Land use Intensity-restricted Multi-objective Spatial Optimization (LIr-MSO) model with more realistic patch size initialization, novel mutation, elite strategies, and objectives balanced via nominalizations and weightings. We tested the model for Dapeng, China where experiments compared comprehensive fitness (across conversion cost, Gross Domestic Product (GDP), ecosystem services value, compactness, and conflict degree) with three contrast experiments, in which changes were separately made in the initialization and mutation. The comprehensive model gave superior fitness compared to the contrast experiments. Iterations progressed rapidly to near-optimality, but final convergence involved much slower parent–offspring mutations. Tradeoffs between conversion cost and compactness were strongest, and conflict degree improved in part as an emergent property of the spatial social connectedness built into our algorithm. Observations of rapid iteration to near-optimality with our model can facilitate interactive simulations, not possible with current models, involving land-use planners and regional managers.

Optimization and Analysis of Plastic Film Consumption for Wrapping Round Baled Silage Using Combined 3D Method Considering Effects of Bale Dimensions

HighlightsSolved the problem of optimal design, in the sense of minimal film usage, of round bale diameter and height.Necessary and sufficient optimality conditions derived in the form of easy-to-solve cubic equations.Bales of a maximum volume achievable with an actual wrapper and of optimal dimensions ensure minimal film usage.Up to 10% savings in film usage if bale dimensions are optimally designed and wrapping parameters properly selected.Abstract. The combined 3D method is used for wrapping cylindrical bales of agricultural materials based on biaxial rotation of the film applicators. The demand for minimization of plastic film consumption keeps increasing, with the goal to save the environment, reduce plastic costs, and minimize wrapping time. Consequently, methods have been reported to solve the problem of optimal wrapping parameters for the conventional wrapping method. In this article, a model-based problem of such a design based on round bale dimensions (diameter and height) that minimizes film consumption for the combined 3D method is mathematically formulated and analyzed. The film consumption per unit of bale volume is used as a measure of film usage. Generally, it is difficult to find the optimal bale dimensions that minimize the original film usage index, due to the discontinuity of the index. Thus, near-optimal parameters, being as important as optimal parameters for engineering applications, are looked for. The problem of selecting near-optimal bale dimensions was constructed by minimizing the continuous lower bound of the original film usage index. The necessary and sufficient optimality conditions for near-optimal bale dimensions were established in the form of standard cubic equations, which can easily be solved using both analytical and numerical methods. Based on the optimality conditions, analytical and numerical analyses were performed of the influence of film width, pre-assumed bale volume, and numbers of bottom and upper film layers on the near-optimal bale dimensions and film usage. The results indicated that the near-optimal bale diameter and height, hereinafter called optimal, monotonically increase, while the optimal film consumption monotonically decreases, with increasing pre-assumed bale volume. Therefore, it is recommended to use bales of a maximum volume achievable with an actual wrapper and of optimal dimensions, i.e., diameter and height. The film width also influences the optimal bale dimensions and film usage: the wider the film, the smaller the minimal film usage. To confirm the effect of near-optimal bale design on film usage, the errors of the near-optimality were examined for four to sixteen film layers. The results of the numerical experiments demonstrated that for four to sixteen layers of film, there are compositions of the bottom and upper film layers for which the relative near-optimality errors do not exceed 0.01% whenever the optimal bale dimensions are used. Simultaneously, inappropriate selection of wrapping parameters may result in increased film usage, measured by mean relative errors of 1% to 9.5%, which means up to 10% film cost savings when the bale dimensions are optimally designed according to the proposed approach, and the wrapping parameters are appropriately selected. Keywords: 3D bale wrapping, Mathematical model, Minimal film consumption, Round bales, Stretch film usage.

Near-optimal control and threshold behavior of an avian influenza model with spatial diffusion on complex networks

<abstract><p>Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns the near-optimal control of an avian influenza model with saturation on heterogeneous complex networks. Firstly, the basic reproduction number $ \mathcal{R}_{0} $ is defined for the model, which can be used to govern the threshold dynamics of influenza disease. Secondly, the near-optimal control problem was formulated by slaughtering poultry and treating infected humans while keeping the loss and cost to a minimum. Thanks to the maximum condition of the Hamiltonian function and the Ekeland's variational principle, we establish both necessary and sufficient conditions for the near-optimality by several delicate estimates for the state and adjoint processes. Finally, a number of examples presented to illustrate our theoretical results.</p></abstract>