Structural anti-optimization with interval design parameters

The anti-optimization problem of structures with uncertain design variables is studied by combing the conventional optimization and interval analysis. The uncertain design parameters, which usually exist in the object function and constraint conditions, are modeled as interval sets. The proposed method can endure the variation of structural performance resulting from the variation of uncertain design parameters. According to the variation range of them, the range or interval of the optimal objective function and the optimal solution can be determined. In this sense, the optimal solution is one domain rather than a point. Numerical examples are used to illustrate the feasibility and superiority of the non-probabilistic optimization method in comparison with the conventional and probabilistic optimization methods.     

Optimal design of continuous time irrational filter with a set of fractional order gammatone components via norm relaxed sequential quadratic programming approach

This paper proposes a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components. The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem. The nonsmooth function is approximated by a smooth operator. The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem. To find a near globally optimal solution, the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem. The current or the previous locally optimal solutions are kicked out by adding the random vectors to them. The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded. By iterating the above procedures, a near globally optimal solution is found. The designed filter is applied to perform the denoising. It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method, while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method. Also, the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters.     

OPTIMAL FEEDER RECONFIGURATION FOR LOSS REDUCTION BY ROBUST SEARCHING HYBRID DIFFERENTIAL EVOLUTION

This paper introduces a robust searching hybrid differential evolution (RSHDE) method to solve the optimal feeder reconfiguration for power loss reduction. The feeder reconfiguration of distribution systems is to recognize beneficially load transfers so that the objective function composed of power losses is minimized and the prescribed voltage limits are satisfied. Mathematically, the problem of this research is a nonlinear programming problem with integer variables. This paper presents a new approach, which uses the RSHDE algorithm with integer variables to solve the problem. Owing to handle the integer variables, the HDE may fail to find the initial search direction for large-scale integer system. This is because the HDE applies a random search at its initial stages. Therefore, two new schemes, the multidirection search scheme and the search space reduction scheme, are embeded into the HDE. These two schemes are used to enhance the search ability before performing the initialization step of the solution process. One three-feeder distribution system from the literature and one practical distribution network of Taiwan Power Company (TPC) are used to exemplify the performance of the proposed method. Moreover, the previous HDE, simulated annealing (SA) and genetic algorithms (GA) methods are also applied to the same example systems for the purpose of comparison. Numerical results show that the proposed method is better than the other methods.     

A hybrid genetic algorithm for constrained multi-objective optimization under uncertainty and target matching problems

This work presents a new approach for interval-based uncertainty analysis. The proposed approach integrates a local search strategy as the worst-case-scenario technique of anti-optimization with a constrained multi-objective genetic algorithm. Anti-optimization is a term for an approach to safety factors in engineering structures which is described as pessimistic and searching for least favorable responses, in combination with optimization techniques but in contrast to probabilistic approaches. The algorithm is applied and evaluated to be efficient and effective in producing good results via target matching problems: a simulated topology and shape optimization problem where a ‘target’ geometry set is predefined as the Pareto optimal solution and a constrained multiobjective optimization problem formulated such that the design solutions will evolve and converge towards the target geometry set.     

Structural design under bounded uncertainty—Optimization with anti-optimization

In many cases precise probabilistic data are not available on uncertainty in loads, but the magnitude of the uncertainty can be bound. This paper proposes a design approach for structural optimization with uncertain but bounded loads. The problem of identifying critical loads is formulated mathematically as an optimization problem in itself (called anti-optimization), so that the design problem is formulated as a two-level optimization. For linear structural analysis it is shown that the antioptimization part is limited to consideration of the vertices of the load-uncertainty domain. An example of a ten-bar truss is used to demonstrate that we cannot replace the anti-optimization process by considering the largest possible loads.     

Optimal nonuniform sampling interval and test-input design for identification of physiological systems from very limited data

Optimal design of test-inputs and sampling intervals in experiments for linear system identification is treated as a nonlinear integer optimization problem. The criterion is a function of the Fisher information matrix, the inverse of which gives a lower bound for the covariance matrix of the parameter estimates. Emphasis is placed on optimum design of nonuniform data sampling intervals when experimental constraints allow only a limited number of discrete-time measurements of the output. A solution algorithm based on a steepest descent strategy is developed and applied to the design of a biologic experiment for estimating the parameters of a model of the dynamics of thyroid hormone metabolism. The effects on parameter accuracy of different model representations are demonstrated numerically, a canonical representation yielding far poorer accuracies than the original process model for nonoptimal sampling schedules, but comparable accuracies when these schedules are optimized. Several objective functions for optimization are compared. The overall results indicate that sampling schedule optimization is a very fruitful approach to maximizing expected parameter estimation accuracies when the sample size is small.     

Probing signals for output constrained processes

Probing input signal design for enhancing the identification of process parameters is developed for scalar input-output processes with output constraints. The sensitivity approach is used and the problem is formulated as an open loop optimal control problem. The solution is via integer programming. Numerical examples illustrate the results.     

An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems

Stochastic model predictive control hinges on the online solution of a stochastic optimal control problem. This paper presents a computationally efficient solution method for stochastic optimal control for nonlinear systems subject to (time‐varying) stochastic disturbances and (time‐invariant) probabilistic model uncertainty in initial conditions and parameters. To this end, new methods are presented for joint propagation of time‐varying and time‐invariant probabilistic uncertainty and the nonconservative approximation of joint chance constraint (JCC) on the system state. The proposed uncertainty propagation method relies on generalized polynomial chaos and conditional probability rules to obtain tractable expressions for the state mean and covariance matrix. A moment‐based surrogate is presented for JCC approximation to circumvent construction of the full probability distribution of the state or the use of integer variables as required when using the sample average approximation. The proposed solution method for stochastic optimal control is illustrated on a nonlinear semibatch reactor case study in the presence of probabilistic model uncertainty and stochastic disturbances. It is shown that the proposed solution method is significantly superior to a standard random sampling method for stochastic optimal control in terms of computational requirements. Furthermore, the moment‐based surrogate for the JCC is shown to be substantially less conservative than the widely used distributionally robust Cantelli‐Chebyshev inequality for chance constraint approximation.     

Stochastic optimal open-loop feedback control: Approximate solution of the Hamiltonian system

Considering a dynamic control system with random model parameters and using the stochastic Hamilton approach stochastic open-loop feedback controls can be determined by solving a two-point boundary value problem (BVP) that describes the optimal state and costate trajectory. In general an analytical solution of the BVP cannot be found. This paper presents two approaches for approximate solutions, each consisting of two independent approximation stages. One stage consists of an iteration process with linearized BVPs that will terminate when the optimal trajectories are represented. These linearized BVPs are then solved by either approximation fixed-point equations (first approach) or Taylor-Expansions in the underlying stochastic model parameters (second approach). This approximation results in a deterministic linear BVP, which can be handled by solving a matrix Riccati differential equation.     

Minimax estimation by probabilistic criterion

A minimax estimation problem in multidimensional linear regression model containing uncertain parameters and random quantities is considered. Simultaneous distribution of random quantities that are a part of the observation model is not prescribed exactly; however, it has a fixed mean and a covariance matrix from the given set. For estimation algorithm optimization, we applied a minimax approach with the risk measure in the form of the exceedance probability of the estimate of a prescribed level by an error. It was shown that a linear estimation problem is equivalent to the minimax problem with the mean-square criterion. In addition, the corresponding linear estimate will be the best (in the minimax sense) by the probabilistic criterion at the class of all unbiased estimates. The least favorable distribution of random model parameters is also constructed. Several partial cases and a numerical example are considered.     

A MIXED INTEGER DYNAMIC PROGRAMMING APPROACH TO A CLASS OF OPTIMAL CONTROL PROBLEMS IN HYBRID SYSTEMS

An optimal control problem for hybrid systems is formulated based on hybrid machine models. A practical approach to solving the problem, suitable for a class of hybrid systems, is presented. This approach consists of transforming the hybrid machine model into a dynamic programming model. Transition costs, in this latter model, are computed using mixed integer programs formulated based on the structure of the original hybrid machine. It is shown that the optimal solution for this dynamic program corresponds to the optimal control decision sequence in the hybrid system. Practical examples, inspired from process-oriented industry applications, are provided to illustrate the solution approach.     

Searching for optimal integer solutions to set partitioning problems using column generation

In this paper, we describe a new approach to increase the possibility of finding integer feasible columns to a set partitioning problem (SPP) directly in solving the linear programming (LP) relaxation using column generation. Traditionally, column generation is aimed to solve the LP‐relaxation as quickly as possible without any concern for the integer properties of the columns formed. In our approach, we aim to generate columns forming an optimal integer solution while simultaneously solving the LP‐relaxation. Using this approach, we can improve the possibility of finding integer solutions by heuristics at each node in the branch‐and‐bound search. In addition, we improve the possibility of finding high‐quality integer solutions in cases where only the columns in the root node are used to solve the problem. The basis of our approach is a subgradient technique applied to a Lagrangian dual formulation of the SPP extended with an additional surrogate constraint. This extra constraint is not relaxed and is used to better control the subgradient evaluations and how the multiplier values are computed. The column generation is then directed, via the multipliers, to construct columns that form feasible integer solutions. Computational experiments show that we can generate optimal integer columns in a large set of well‐known test problems as compared to both standard and stabilized column generation, and simultaneously keep the number of columns smaller than standard column generation. This is also supported by tests on a case study with work‐shift generation.     

随机批量物料搬运的制造系统AGV配置优化

针对定制型装备制造企业智能车间中物料搬运系统的AGV数量配置问题,以最小化AGV投资成本为目标,建立具有系统产能和订单交货期双重约束的优化模型.由于该优化问题是一个随机非线性的整数规划问题,且约束条件无法用决策变量的封闭形式表示,提出一种基于排队网求解性能指标值的禁忌搜索算法求解该问题.基于马尔可夫理论,提出拓展的状态...     

Heuristic solutions to the long-term unit commitment problem with cogeneration plants

We consider a long-term version of the unit commitment problem that spans over one year divided into hourly time intervals. It includes constraints on electricity and heating production as well as on biomass consumption. The problem is of interest for scenario analysis in long-term strategic planning. We model the problem as a large mixed integer programming problem. Two solutions to this problem are of interest but computationally intractable: the optimal solution and the solution derived by market simulation. To achieve good and fast approximations to these two solutions, we design heuristic algorithms, including mixed integer programming heuristics, construction heuristics and local search procedures. Two setups are the best: a relax and fix mixed integer programming approach with an objective function reformulation and a combination of a dispatching heuristic with stochastic local search. The work is developed in the context of the Danish electricity market and the computational analysis is carried out on real-life data.     

Optimal Path Problems with Second-Order Stochastic Dominance Constraints

This paper studies optimal path problems integrated with the concept of second order stochastic dominance. These problems arise from applications where travelers are concerned with the trade off between the risks associated with random travel time and other travel costs. Risk-averse behavior is embedded by requiring the random travel times on the optimal paths to stochastically dominate that on a benchmark path in the second order. A general linear operating cost is introduced to combine link- and path-based costs. The latter, which is the focus of the paper, is employed to address schedule costs pertinent to late and early arrival. An equivalent integer program to the problem is constructed by transforming the stochastic dominance constraint into a finite number of linear constraints. The problem is solved using both off-the-shelf solvers and specialized algorithms based on dynamic programming (DP). Although neither approach ensures satisfactory performance for general large-scale problems, the numerical experiments indicate that the DP-based approach provides a computationally feasible option to solve medium-size instances (networks with several thousand links) when correlations among random link travel times can be ignored.     

Algorithms,software and sequential optimal sampling schedule designs for pharmacokinetic and physiologic experiments

The problem of designing optimal blood sampling protocols for kinetic experiments in pharmacology, physiology and medicine is briefly described, followed by a presentation of several interesting results based on sequentially optimized studies we have performed in more than 75 laboratory animals. Experiences with different algorithms and design software are also presented. The overall approach appears to be highly efficacious, from the standpoints of both laboratory economics and resulting model accuracy. Optimal sampling schedules (OSS) have a number of different time points equal to the number of unknown parameters for a popular class of models. Replication rather than distribution of samples provide maximum accuracy when additional sampling is feasible; and specific replicates can be used to weight some parameter accuracies more than others, even when a D-optimality criterion is used. Our sequential experiment scheme often converged in 1 step and resulting optimal sampling schedules were reasonably robust, allowing for biological variation among the animals studied.     

Integer approximation of 3D chamfer mask coefficients using a scaling factor in anisotropic grids

For discrete data represented by three variable scalar functions, the sampling step may be different according to the axes, leading to parallelepipedic sampling grids. This is the case for instance with medical or industrial computed tomography and confocal microscopy, as well as in grey level image analysis if images are modelized by means of their set representation (mathematical morphology). In this paper, 3D non-cubic chamfer masks are introduced. The problem of coefficient optimization is addressed for arbitrary mask size. Thank to this, first, the maximal normalized error with respect to Euclidean distance can be derived analytically, in any 3D anisotropic lattice, and second, optimal chamfer mask coefficients can be computed. We propose a method to calculate lower and upper bounds for integer scaling factors in order to obtain integer approximations for the coefficients. This approach helps the algorithm perform in scenarios where memory is limited.     

Sampling of piecewise constant stochastic processes with smooth distributions of stay time in states

In the solution of sampling problem, a requirement is that the probability of skipping a state must not exceed a prescribed value. A general solution is obtained that makes it possible to construct approximate procedures of various accuracy. Formulas for the Erlang distribution of the stay times and formulas based on the recalculation of the first two moments are obtained. Computed characteristics are presented and an illustrative example is discussed.     

A naive Bayes probability estimation model based on self-adaptive differential evolution

In the process of learning the naive Bayes, estimating probabilities from a given set of training samples is crucial. However, when the training samples are not adequate, probability estimation method will inevitably suffer from the zero-frequency problem. To avoid this problem, Laplace-estimate and M-estimate are the two main methods used to estimate probabilities. The estimation of two important parameters m (integer variable) and p (probability variable) in these methods has a direct impact on the underlying experimental results. In this paper, we study the existing probability estimation methods and carry out a parameter Cross-test by experimentally analyzing the performance of M-estimate with different settings for the two parameters m and p. This part of experimental result shows that the optimal parameter values vary corresponding to different data sets. Motivated by these analysis results, we propose an estimation model based on self-adaptive differential evolution. Then we propose an approach to calculate the optimal m and p value for each conditional probability to avoid the zero-frequency problem. We experimentally test our approach in terms of classification accuracy using the 36 benchmark machine learning repository data sets, and compare it to a naive Bayes with Laplace-estimate and M-estimate with a variety of setting of parameters from literature and those possible optimal settings via our experimental analysis. The experimental results show that the estimation model is efficient and our proposed approach significantly outperforms the traditional probability estimation approaches especially for large data sets (large number of instances and attributes).     

Hybrid approach for the two‐dimensional bin packing problem with two‐staged patterns

This paper presents a two‐phase heuristic approach for the two‐dimensional bin packing problem with two‐staged patterns and nonoriented items. A solution is generated in each phase and the better one is selected. Residual problems are solved by column generation in the first phase, where a partial admitting procedure is used to admit some of the patterns into the phase‐1 solution. The second solution is obtained from solving an integer linear programming problem over the set of all patterns generated in the first phase, where a time limit is used and subsequently the solution may not be optimal over the pattern set. The computational results indicate that the approach yields the best solution quality among the heuristics that use two‐staged or more complex patterns.