Uncertain Objective Function Research Articles

Constructing order-2 information granules of linguistic expressions with the aid of the principle of justifiable granularity

To capture collective opinions/evaluations in a collection of individual linguistic expressions, this study proposes an approach to construct order-2 information granules by extending the numerical data-based principles of justifiable granularity to a linguistic data-based one. First, the two key criteria of the principle of justifiable granularity, namely coverage and specificity, are formally defined in the context of order-2 information granules. Second, three order-2 information granules construction models by maximizing the product of coverage and specificity are developed for coping with one-dimensional direct linguistic expressions, linguistic preference relations, and multi-dimensional direct linguistic expressions, respectively. Third, considering that the developed order-2 information granules construction models exhibit a non-linear uncertain objective function and equality constraints, the constrained multi-swarm PSO without velocity is improved with the use of the ε constrained handling technique for effectively solving the models. Case studies on the considered three types of linguistic expressions show the applicability of the proposed models and ensuing algorithm. The superiority of the improved algorithm in handling the proposed models is demonstrated by comparison with the original one. The effectiveness of the proposed models in terms of abnormal corrective ability and balance of coverage and specificity is verified by comparing with a family of Top-n methods. The originality of this study lies in the construction of an operational entity, namely order-2 information granule, that reflects the group opinion without specifying the formalism of individual linguistic expressions, which provides an effective and efficient way to aggregate ubiquitous linguistic information for subsequent computation, reasoning and decision-making.

Uncertain optimization for composite base plate of mortar under impact load

Information warfare places higher requirements on the remoteness, precision and mobility of artillery weapon systems. As an indispensable and important category of artillery, mortar plays an important role. As an important component of mortar, the base plate bears tremendous impact load. Its design level has an important influence on the shooting stability, shooting accuracy and maneuverability of the mortar. Composite base plate helps improve the maneuverability of the mortar, but it will also make the force and movement rule of mortar launching more complex and lead to a more acute contradiction between artillery power and maneuverability, which propose a huge challenge to modern artillery optimization design theory. Under this background, this manuscript conducts research on the structural uncertain optimization of composite material base plate for the mortar. In order to obtain the optimal solution and reasonable interval of uncertain parameters with main dimension parameters, this manuscript uses interval model to carry out deterministic transformation for uncertain objective function and uncertain constraints, combines multi-objective genetic algorithm to solve the problem, and carries out numerical simulation to verify the optimal value. The result shows that the design idea and method are feasible for uncertain optimization of composite base sheet of mortar under impact load, and the optimization objectives and strategies are effective, which can provide reference for the uncertain optimization research of composite base plate.

An interval uncertainty optimization algorithm based on radial basis function network differentiation

A new interval uncertainty optimization algorithm is proposed to replace two-layer nested optimization, owing to the low efficiency of the latter. The radial basis function network is established to obtain the first-order differential, which is difficult to achieve in practical engineering problems. The results obtained by this network differential method are verified by a mathematical example. The network differential method is combined with the interval perturbation method to compute the bounds of uncertain objective functions and constraints, and the subinterval method is introduced to address the large level of uncertainty. The example of a compression spring shows the feasibility of this interval analysis method. The interval uncertain optimization problem is transformed into a deterministic one through the interval order relationship and probability model, and solved using the genetic algorithm or non-dominated sorting genetic algorithm-II. A numerical example and electromagnetic buffer model demonstrate the accuracy, efficiency and practicability of this new method.

Robust optimization with belief functions

In this paper, an optimization problem with uncertain objective function coefficients is considered. The uncertainty is specified by providing a discrete scenario set containing possible realizations of the objective function coefficients. The concept of belief function in the traditional and possibilistic setting is applied to define a set of admissible probability distributions over the scenario set. The generalized Hurwicz criterion is then used to compute a solution. In this paper, the complexity of the resulting problem is explored. Some exact and approximation methods of solving it are proposed.

Robust optimality analysis for linear programming problems with uncertain objective function coefficients: an outer approximation approach

Robust optimality analysis for linear programming problems with uncertain objective function coefficients: an outer approximation approach

Eco-driving at signalized intersections under uncertain traffic conditions

Eco-driving at signalized intersections is a topical issue in CAVs technology and an important part of energy-efficient driving. In this paper, a robust eco-driving control method of vehicular platoon is proposed to address the uncertain traffic conditions, including traffic queuing effects and signal time variations. Firstly, an effective signalization model is established by estimating the traffic queue length and describing the uncertain parameters with the interval number. Then, taking fuel economy and vehicle safety as the objectives, the uncertain objective function is transformed into a deterministic optimization problem by using the order interval relation model. Finally, a model predictive control method is used to optimize the problem. The simulation results indicate that the proposed method can significantly improve the control robustness and fuel economy without the signal history data.

A robust, multi-solution framework for well placement and control optimization

Field development and control optimization aim to maximize the economic profit of oil and gas production while considering several sources of uncertainty. This results in a high-dimensional optimization problem with a computationally demanding and uncertain objective function based on the simulated reservoir model. The limitations of many current robust optimization methods are: 1) it is single-level optimization (e.g. optimization of well locations/placement only; or of well production/injection control variables only) that ignores interference between the control variables from different levels; and 2) they provide a single optimal solution, whereas operational problems often add unexpected constraints likely to reduce that optimal, inflexible solution to a sub-optimal scenario. This paper presents a robust, multi-solution framework based on sequential iterative optimization of control variables at multiple levels using the Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm. A systematic realization selection process, tailored to the objective of the subsequent optimization stage, is used to select a small representative ensemble of reservoir model realizations to be used for calculating the expected objective value. The estimated gradients are calculated using a 1:1 ratio mapping ensemble of control variables perturbations at each iteration onto the ensemble of selected reservoir model realizations to reduce the computational cost. An ensemble of close-to-optimum solutions is then chosen from each level (e.g. from the well placement optimization level) and transferred to the next level of optimization (e.g. where the control settings are optimized), and this loop continues until no significant improvement is observed in the expected objective value. Fit-for-purpose clustering techniques are developed to systematically select an ensemble of solutions, with maximum differences in control variables but close-to-optimum objective values, at each optimization level. The proposed framework has been tested on a benchmark case study (Brugge field). Multiple solutions are obtained with different well locations and control settings but close-to-optimum objective values. We show that suboptimal solutions from an early optimization level can approach and even outdo the optimal one at the next level(s). Results demonstrate the advantage of the developed framework in more efficient exploration of the search space and providing the much-needed operational flexibility to field operators.

Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints

This work proposes a Bayesian optimization (BO) method for solving multi-objective robust design optimization (RDO) problems of steel frames under aleatory uncertainty in external loads and material properties. Joint and individual probabilistic constrained RDO problems are formulated to consider two different ways the frame reaches its collapse state. Each problem involves three conflicting objective functions, namely, the total mass of the frame, the mean and variance of the maximum inter-story drift. Since the uncertain objective and probabilistic constraint functions of both problems are implicit within a finite element analysis program and the computation of the probabilistic constraints is an NP-hard problem, BO is used to guide the optimization process toward better solutions after it completes an iteration and offers a set of near Pareto-optimal solutions when it terminates. Specifically, Bayesian regression models called Gaussian processes (GPs) serve as surrogates for the structural responses. Two acquisition functions are then developed for the two RDO problems and a maximization problem of these functions is formulated as a mixed-integer nonlinear programming (MINLP) problem. A new random search coupled with simulated annealing is devised to solve the MINLP problem, thereby locating the most promising point in the input variable space at which the current solutions maximize their chance to be improved and the GP models are refined before the BO starts a new iteration. A test problem and two design examples show that exact or good Pareto-optimal solutions to the RDO problems can be found by the proposed method with 20 iterations.

An efficient nonlinear interval uncertain optimization method using Legendre polynomial chaos expansion

Many specialized numerical techniques have been developed on interval uncertainty optimization. However, the conventional nested optimization strategy is time-consuming. The improved Taylor-based interval optimization avoids the laborious inner optimization, nonetheless, it is intrusive and presents significant computational challenges in the large-level uncertain optimization problems. To end these, this paper proposes a Legendre polynomial expansion method combined with the subinterval technique to evaluate the range enclosure of an interval function, where the expansion coefficients are computed through the collocation method. The detailed comparisons demonstrate that such Legendre-based polynomial expansion provides much more confidence in interval bounds evaluation than the Taylor expansion, and is capable of maintaining very sharp solution enclosures even when the uncertainty level is increased. More exciting, it is highly efficient. Subsequent application of the Legendre polynomial expansion in the interval bound estimation of the uncertain objective function and constraints, and combining an appropriate outer optimizer, result in a novel nonlinear interval uncertain optimization method for complicated engineering systems, whereby the time-consuming optimization nesting is avoided. A beam design example and an electromagnetic buffer optimization example illustrate that the proposed interval optimization method has fine computational precision and high computational efficiency, especially is capable of attacking the increase of uncertainty level. Moreover, the new method does not require the derivative information of the uncertain objective and constraints, such that applies to any engineering uncertain optimization problems, including the complex engineering problems which are usually referred to as “black-box”.

A Bayesian approach for quantile optimization problems with high-dimensional uncertainty sources

Robust optimization strategies typically aim at minimizing some statistics of the uncertain objective function and can be expensive to solve when the statistic is costly to estimate at each design point. Surrogate models of the uncertain objective function can be used to reduce this computational cost. However, such surrogate approaches classically require a low-dimensional parametrization of the uncertainties, limiting their applicability. This work concentrates on the minimization of the quantile and the direct construction of a quantile regression model over the design space, from a limited number of training samples. A Bayesian quantile regression procedure is employed to construct the full posterior distribution of the quantile model. Sampling this distribution, we can assess the estimation error and adjust the complexity of the regression model to the available data. The Bayesian regression is embedded in a Bayesian optimization procedure, which generates sequentially new samples to improve the determination of the minimum of the quantile. Specifically, the sample infill strategy uses optimal points of a sample set of the quantile estimator. The optimization method is tested on simple analytical functions to demonstrate its convergence to the global optimum. The robust design of an airfoil’s shock control bump under high-dimensional geometrical and operational uncertainties serves to demonstrate the capability of the method to handle problems with industrial relevance. Finally, we provide recommendations for future developments and improvements of the method.

Interval Method Based Optimal Scheduling of Regional Multi-Microgrids With Uncertainties of Renewable Energy

In order to face the challenges of high proportion of wind and photovoltaic in regional multi-microgrids, a coordinated interval optimization scheduling method considering uncertainties of renewable energy is proposed. The interval optimization is used to describe the uncertainties of wind power and photovoltaic through interval numbers, then an interval optimization model for a regional multi-microgrid coordinated operation system including the uncertain objective function and constraints is established, based on the interval order relationship and possibility, the uncertain objective function and constraints can be transformed into deterministic ones. The different micro-source outputs and operating costs are compared under the independent and coordinated operation mode. It is verified that the optimization of the coordinated operation interval of the multi-microgrid can effectively reduce the operating cost of the system and stabilize the load fluctuation. The interval optimization method describes the wind power and photovoltaic as fluctuation intervals, and the corresponding operating cost intervals are obtained with different uncertainties, the scheduling results can more truly reflect the impact of uncertainty on the system operation results. At the same time, this method can provide system dispatchers with information about the influence of wind power and photovoltaic uncertainty on the system optimization results, which is convenient for dispatchers to make decisions.

An interval robust design optimization method and its application in heat transfer problems

ABSTRACT An interval robust optimization method is suggested to improve the robustness and performance of the process or product in design. A general uncertain robust optimization problem is considered in which the objective function and constraints are both nonlinear and uncertain, and the uncertainties of design variables and uncertain parameters are all included and depicted by the interval model. A nonlinear interval number programming method is introduced to solve the robust optimization model, in which the uncertain objective function is described by a pair of deterministic functions, and the uncertain constraints are converted to deterministic constraints on the basis of the reliability-based possibility degree of interval. The method is then applied to several heat transfer problems for performance optimization of the heat transfer processes while guaranteeing the robustness of the optimal design.

Research on Multi-Alternatives Problem of Finite Element Model Updating Based on IAFSA and Kriging Model.

Due to insufficient test data, insufficient constraint equations and uncertain objective function, the local optimal solution and the global optimal solution of the objective function in finite element model updating may represent the actual parameters of the structure. Based on this, this paper proposes an improved artificial fish school algorithm. By combining the niche technology with the artificial fish school algorithm, the improved algorithm can systematically find multiple global optimal solutions and local optimal solutions of the objective function. Aiming at the difficulty of determining the niche radius, an adaptive niche radius mechanism is proposed. The improved algorithm is used to study the multi-alternatives problem of finite element model updating after verifying its feasibility through numerical simulation analysis. In the case of benchmark framework model updating, it is confirmed that multi-alternative problems exist and the global optimal solution of the objective function does not necessarily represent the true parameters of the structure. In case 2, the improved algorithm combined with the Kriging model is applied to the model updating of a cable-stayed footbridge, and 15 sets of solutions are obtained, in which the error objective function values of the measured and theoretical values of the bridge modes are close but the solutions are completely different. Combining with the actual bridge condition and reanalysis technology, the author takes the suboptimal solution 2 as the most representative solution of the bridge parameters, which reduces the possibility of misjudgment of structural parameters.

A note on [formula omitted]-completeness of a robust binary linear program with binary uncertainty set

We show that the robust binary linear program with uncertain objective function and binary uncertainty set defined by linear constraints is Σ2p-complete. This result is then extended to the K-adaptability version of the problem, where one of K pre-calculated solutions can be chosen after the uncertain parameter is revealed.

Hurwicz model of uncertain optimal control with jump

How to choose the optimization criterion of the objective function is an important issue for uncertain optimal control. The Hurwicz criterion is a flexible optimization criterion attempting to find the intermediate area between the extremes posed by the optimistic and pessimistic criteria. Based on uncertainty theory, in this paper, we establish a new uncertain optimal control model with jump by making use of Hurwicz criterion to optimize an uncertain objective function. By applying Bellman's principle of optimality, the principle of optimality for the proposed model is presented and then the equation of optimality is derived. Finally, an example is given to show the the effectiveness of the results obtained.

An interval uncertain optimization method using back-propagation neural network differentiation

An interval uncertain optimization method is proposed for engineering uncertain optimization problems. The first-order derivative information may be unavailable in practical engineering systems. To overcome this drawback, the method to calculate the first-order partial derivatives with a back-propagation neural network is investigated, and a numerical example is presented to demonstrate its fine precision. In the optimization process, the back-propagation neural network and its first-order derivatives are introduced into the interval perturbation method and combined with the subinterval method, the extrema of the uncertain objective function and uncertain constraints can be approximated without much computational cost. Thus the most challenging problem in current interval optimization, namely low optimization efficiency caused by nested optimization can be solved to a certain extent. A two-bar structure system is adopted to validate its fine precision and high computational efficiency in interval bounds estimation. Based on the interval order relation and interval possibility degree model, an interval uncertain optimization model can be transformed into a deterministic optimization model, and be solved by some auxiliary optimization algorithm. A numerical example and an engineering application have indicated that this method is efficient for uncertain optimization problems.

A class of multi-parametric quadratic program with an uncertain objective function

In this paper we analyze a class of multi-parametric quadratic program (mpQP) with parameters in the objective function. Except for parameters in coefficients associated with the linear term, the coefficient of the quadratic term, which is a positive definite matrix, is multiplied by a scalar parameter, while the quadratic coefficient of a standard mpQP is deterministic. We reveal the optimal solution is a linear fractional function in the parameters, and the critical regions remain polyhedral. The discussed mpQP can be reformulated as a standard mpQP via variable and parameter transformations. The proposed method is used to evaluate the economic operation of a residential energy system under time-and-level-of-use electricity pricing, highlighting the potential application in practical problems.

Towards Efficient Robust Optimization using Data based Optimal Segmentation of Uncertain Space

Performing multi-objective optimization under uncertainty is a common requirement in industries and academia. Robust optimization (RO) is considered as an efficient and tractable approach provided one has access to behavioral data for the uncertain parameters. However, solutions of RO may be far from the real solution and less reliable due to inability to map the uncertain space accurately, especially when the data appears discontinuous and scattered in the uncertain domain. Amalgamating machine learning algorithms with RO, this paper proposes a data-driven methodology, where a novel fuzzy clustering mechanism is implemented along-with boundary construction, to transcript the uncertain space such that the specific regions of uncertainty are identified. Subsequently, using intelligent Sobol sampling, samples are generated in the mapped uncertain regions. Results of two test cases are presented along with a comprehensive comparison study. Considered case-studies include highly nonlinear model for continuous casting process from steelmaking industries, where a multi-objective optimization problem under uncertainty is solved to balance the conflict between productivity and energy consumption. The Pareto-optimal solutions of the resulting RO problem are obtained through Non-Dominated Sorting Genetic Algorithm – II, and ~23–29% improvement is observed in the uncertain objective function. Further, the spread and diversity metrics are enhanced by ~10–95% as compared to those obtained using other standard uncertainty sets.

Uncertain optimization of in-bore launching performance of artillery based on interval method

In order to study the uncertainty optimization of the in-bore launching performance of larger-caliber artillery, the interior ballistic program was compiled and embedded in ABAQUS finite element software for secondary development, and the dynamic model of in-bore launching was built. The structural parameters of bore, the structural parameters of projectile and parameters of launching propellant were considered, and the uncertainty was described by interval number. BP neural network was used to construct a surrogate model for the dynamic model of in-bore launching. The deterministic transformation of uncertain objective function and uncertain constraints was carried out by using interval order model and interval probability model respectively. The interval uncertainty optimization model of the in-bore launching performance artillery was established by taking the interval radius and midpoint of the projectile muzzle velocity as objective functions and the maximum chamber pressure as constraint. Multi-objective genetic algorithm was used to solve the problem, and the optimal solution and reasonable interval of uncertain parameters were obtained.

K-adaptability in stochastic combinatorial optimization under objective uncertainty

Abstract We address combinatorial optimization problems with uncertain objective functions, given by discrete probability distributions. Within this setting, we investigate the so-called K-adaptability approach: the aim is to calculate a set of K feasible solutions such that the objective value of the best of these solutions, calculated in each scenario independently, is optimal in expectation. Interpreted as a stochastic optimization problem, we only consider second-stage variables, however, the corresponding candidate solutions are selected in the first stage, i.e., before the scenario is known. We show that this problem is NP-hard even if the underlying certain problem is trivial, and present further complexity results concerning approximability and fixed-parameter tractability with respect to K. Moreover, we present exact solution methods as well as a heuristic for this problem and compare them in an extensive experimental evaluation, where the underlying problem is the Unconstrained Binary Optimization Problem, the Shortest Path Problem or the Spanning Tree Problem. It turns out that the performance and the ranking of these approaches strongly depends on the parameter K and on the number of scenarios.