A note on robust 0-1 optimization with uncertain cost coefficients

Spatio-temporal processes are ubiquitous and prevalent across disciplines. Understanding the mechanisms underlying processes and integrating this information into models is of great interest, as it can improve forecasting accuracy and align with scientific motivation. Examples of such models include Partial Differential Equation (PDE) Models or Physics-Informed Neural Network Models in the applied mathematics or deep learning community, respectively. However, these models often overlook uncertainty quantification despite its crucial role, considering that real-world processes necessarily involve inherent errors that physical laws cannot fully explain. Dynamic Spatio-Temporal Models (DSTMs) offer a flexible and effective approach by embedding physics laws within the Bayesian Hierarchical Model (BHM) framework and accounting for dependencies in space and time conditionally. This dissertation explores integrating physics laws while accounting for uncertainty within BHMs and neural network models for complex environmental processes. To start, a novel approach utilizing a level-set method and low-rank representation within a BHM is developed to model the evolution of wildfire boundaries in the presence of uncertainty in data and a lack of knowledge about the boundaries. Subsequently, a hybrid model that nests an echo state network within a level-set method to accommodate nonlinearity is developed. This model is computationally efficient and includes calibrated uncertainty quantification. Lastly, a new class of DSTMs, capable of accommodating both high and low extremes through a regime-switching scheme of stable distributions with varying tail indices, is presented. This last method is illustrated on fine particulate matter (PM2.5) observations emanating from wildfires in the prairie region of the US.

Though accident data have been collected across industries, they may inherently contain uncertainty of randomness and fuzziness which in turn leads to misleading interpretation of the analysis. To handle the issue of uncertainty within accident data, the present work proposes a rough set theory (RST)-based approach to provide rule-based solution to the industry to minimize the number of accidents at work. Using RST and RST-based rule generation algorithm Learning by Example Module: Version 2 (LEM2), 279 important rules are extracted from the accident data obtained from an integrated steel industry to analyze the incident outcomes (injury, near miss and property damage). The results of the proposed methodology explore some of the important findings which are useful for the industry perspective. Therefore, the RST-based approach can be effective and efficient as well because of its potential to produce good results in the presence of uncertainty in data.

A surrogate-assisted evolution strategy for constrained multi-objective optimization

We present a novel method for predicting the performance of an object recognition approach in the presence of data uncertainty, occlusion and clutter. The recognition approach uses a vote-based decision criterion, which selects the object/pose hypothesis that has the maximum number of consistent features (votes) with the scene data. The prediction method determines a fundamental, optimistic, limit on achievable performance by any vote-based recognition system. It captures the structural similarity between model objects, which is a fundamental factor in determining the recognition performance. Given a bound on data uncertainty, we determine the structural similarity between every pair of model objects. This is done by computing the number of consistent features between the two objects as a function of the relative transformation between them. Similarity information is then used, along with statistical models for data distortion, to estimate the probability of correct recognition (PCR) as a function of occlusion and clutter rates. The method is validated by comparing predicted PCR plots with ones that are obtained experimentally.

We propose a new particle swarm optimization algorithm for problems where objective functions are subject to zero-mean, independent, and identically distributed stochastic noise. While particle swarm optimization has been successfully applied to solve many complex deterministic nonlinear optimization problems, straightforward applications of particle swarm optimization to noisy optimization problems are subject to failure because the noise in objective function values can lead the algorithm to incorrectly identify positions as the global/personal best positions. Instead of having the entire swarm follow a global best position based on the sample average of objective function values, the proposed new algorithm works with a set of statistically global best positions that include one or more positions with objective function values that are statistically equivalent, which is achieved using a combination of statistical subset selection and clustering analysis. The new PSO algorithm can be seamlessly integrated with adaptive resampling procedures to enhance the capability of PSO to cope with noisy objective functions. Numerical experiments demonstrate that the new algorithm is able to consistently find better solutions than the canonical particle swarm optimization algorithm in the presence of stochastic noise in objective function values with different resampling procedures.

An indirect search heuristic is described for solving harvest-scheduling problems under adjacency constraints. This method works in combination with a greedy algorithm by diversifying the search through random changes in prioritized harvest queues. The indirect search is tested on a series of tactical problems and compared with published results for tabu search, simulated annealing, integer programming and linear programming. Results for large strategic problems are compared to a simulated annealing search algorithm. Objective function values are comparable to tabu search and simulated annealing, and solution times range from 38 seconds to 40 minutes, depending on the problem size and the number of iterations. Benefits of the indirect search method are: (1) objective function values can be higher than those computed through other heuristic algorithms, and (2) the algorithm produces good results without time-consuming experimentation with parameters of the search algorithm. The method also has potential for solving more complicated, multiple objective problems. FOR. SCI. 49(1):1–11.

A selection hyper-heuristic is used to minimise the objective functions of a well-known set of benchmark problems. The resulting sequences of low level heuristic selections and objective function values are used to generate a database of heuristic selections. The sequences in the database are broken down into subsequences and the mathematical concept of a logarithmic return is used to discriminate between “effective” subsequences, which tend to decrease the objective value, and “disruptive” subsequences, which tend to increase the objective value. These subsequences are then employed in a sequenced based hyper-heuristic and evaluated on an unseen set of benchmark problems. Empirical results demonstrate that the “effective” subsequences perform significantly better than the “disruptive” subsequences across a number of problem domains with 99% confidence. The identification of subsequences of heuristic selections that can be shown to be effective across a number of problems or problem domains could have important implications for the design of future sequence based hyper-heuristics.

In this study, the role of self‐validated computing for solving the energy hub‐scheduling problem in the presence of multiple and heterogeneous sources of data uncertainties is explored and a new solution paradigm based on affine arithmetic is conceptualised. The benefits deriving from the application of this methodology are analysed in details, and several numerical results are presented and discussed.

In this study, we propose a new global optimization method in which the simultaneous perturbation gradient approximation is introduced into a multi-point type chaotic optimization method. The multi-point type chaotic optimization method, which has been proposed recently, is a global optimization method to solve unconstrained optimization problems in which multiple search points which implement global search driven by a chaotic gradient dynamic model are synchronized to their elite search points. The chaotic optimization method uses gradient as a driving force for search points. Hence, its application is confined to a class of problems in which gradient of the objective function can be computed. In this study, we introduce the simultaneous perturbation gradient approximation into the multi-point type chaotic optimization method in order to compute gradient approximately so that the chaotic optimization method can be applied to a class of problems whose objective function values only can be computed. Then, we confirm effectiveness of the proposed method through applications to several unconstrained multi-peaked optimization problems with 100 variables comparing to other major meta-heuristics.

Objective functions in combinatorial optimization are discrete. The number of possible values of a discrete objective function is totally different from problem to problem. Optimization of a discrete objective function is often very difficult. In the case of multiobjective optimization, a different objective function has a different number of possible values. This means that each axis of the objective space has a different granularity. Some axes may have fine granularities while others are coarse. In this paper, we examine the effect of discrete objective functions with different granularities on the search behavior of EMO (evolutionary multiobjective optimization) algorithms through computational experiments. Experimental results show that a discrete objective function with a coarse granularity slows down the search of EMO algorithms along that objective. An interesting observation is that such a slow-down along one objective often leads to the speed-up of the search along other objectives. We also examine the effect of adding a small random noise to each discrete objective function in order to increase the number of possible objective values.

Weakly admissible transformations are introduced for solving algebraic assignment and transportation problems, which cover so important classes as problems with sum objectives, bottleneck objectives, lexicographical objectives and others. A transformation of the cost matrix is called weakly admissible, if there are two constants α and β in the underlying semigroup that for all feasible solutions the composition of α and the objective value with respect to the original cost coefficients is equal to the composition of β and the objective value with respect to the transformed cost coefficients. The elements α and β can be determined by shortest path algorithms. An optimal solution for the algebraic assignment problem can be found after at most n weakly admissible transformations, therefore the proposed method yields an O(n 3) algorithm for algebraic assignment problems.

This article considers the problem of minimizing a real‐valued function defined on the real line (i.e., a single‐dimensional problem). This function is referred to as the objective function . The approach is to use an iterative search algorithm, also called a line‐search method. Single‐dimensional search methods are of interest for the following reasons. First, they are special cases of search methods used in multivariable problems. Second, they are used as part of general multivariable algorithms. An iterative algorithm starts with an initial candidate solution and generate a sequence of iterates . The iterative algorithm may use only values of the objective function, or perhaps also its first derivative and even its second derivative. This article presents several algorithms: the golden section and Fibonacci methods (which use only objective function values), the bisection and secant methods (which use only derivative values), and Newton's method (which uses both first and second derivatives).

In the paper, Location Model of Risk Pooling (LMRP) model based on Compensation and Punishment Policy is built for Logistics Company. The basic objective of the model is to occupy market or reduce the total costs. The model built is a square nonlinear 0–1 integer-programming model which helps to determine which retailer should serve as the distribution center and how to allocate the other retailers to the distribution centers. Applying the particle swarm optimization and using Daskin's computational instances, the suboptimum solution of the model is found and some conclusions are drawn. In a buyer market, logistics corporations should adopt a compensation policy giving a discount price of transportation in order to make up for the time lost to retailers. In a seller market, logistics corporations should adopt punishment policy by charging higher price of transportation in order to reduce distribution costs. By changing distribution radius Dr and compensation or punishment cost factor W, we can draw the following conclusions: Firstly, under compensation policy, compensation cost factor W is positively related with the objective function value and Dr is negatively related with the objective function value. Secondly, under punishment policy, punishment cost factor |W| is negatively related with the objective function value and positively related with the punishment cost and Dr is positively related with the objective function value.

Manufacturing cell formation using an improved P-median model

This work examines the method of analytic centers of Sonnevend when applied to solve generalized convex quadratic programs — where also the constraints are given by convex quadratic functions. We establish the existence of a two-sided ellipsoidal approximation for the set of feasible points around its center and show, that a simple (zero order) algorithm starting from an initial center of the feasible set generates a sequence of strictly feasible points whose objective function values converge to the optimal value. Concerning the speed of convergence it is shown that an upper bound for the gap in between the objective function value and the optimal value is reduced by a factor ofe with\(O(\sqrt m \left| {ln \varepsilon } \right|)\) iterations wherem is the number of inequality constraints. Here, each iteration involves the computation of one Newton step. The bound of\(O(\sqrt m \left| {ln \varepsilon } \right|)\) Newton iterations to guarantee an error reduction by a factor ofe in the objective function is as good as the one currently given forlinear programs. However, the algorithm considered here is of theoretical interest only, full efficiency of the method can only be obtained when accelerating it by some (higher order) extrapolation scheme, see e.g. the work of Jarre, Sonnevend and Stoer.