Nonmonotonic Criterion Research Articles

Lexicographic optimization-based approaches to learning a representative model for multi-criteria sorting with non-monotonic criteria

Deriving a representative model using value function-based methods from the perspective of preference disaggregation has emerged as a prominent and growing topic in multi-criteria sorting (MCS) problems. A noteworthy observation is that many existing approaches to learning a representative model for MCS problems traditionally assume the monotonicity of criteria, which may not always align with the complexities found in real-world MCS scenarios. Consequently, this paper proposes some approaches to learning a representative model for MCS problems with non-monotonic criteria through the integration of the threshold-based value-driven sorting procedure. To do so, we first define some transformation functions to map the marginal values and category thresholds into a UTA-like functional space. Subsequently, we construct constraint sets to model non-monotonic criteria in MCS problems and develop optimization models to check and rectify the inconsistency of the decision maker’s assignment example preference information. By simultaneously considering the complexity and discriminative power of the models, two distinct lexicographic optimization-based approaches are developed to derive a representative model for MCS problems with non-monotonic criteria. Eventually, we offer an illustrative example and conduct comprehensive simulation experiments to elaborate the feasibility and validity of the proposed approaches.

Stochastic Triangular Fuzzy Number (S-TFN) Normalization: A New Approach for Nonmonotonic Normalization

Data normalization is essential in many fields, such as speech recognition, deep learning, machine learning, and optimization. Many researchers focus on developing various normalization techniques, such as min-max, score-based, maximum, vector, or sum, on processing data efficiently. However, classical normalization methods may need to be improved in Multi-Criteria Decision-Analysis (MCDA), where criteria are often non-monotonic. This paper proposes a new approach for identifying and normalizing nonmonotonic criteria in MCDA. The method combines a stochastic optimization technique - a genetic algorithm - with normalization based on triangular fuzzy numbers. This paper compares the proposed approach with classical normalization techniques, such as min-max, maximum, vector, or non-linear, using the classical MCDA method - the Technique of Order Preference Similarity to the Ideal Solution (TOPSIS). The study results showed that the proposed approach to normalizing non-monotonic criteria yields a better match between decision preferences and the actual reference model than traditional normalization methods. Optimization using the genetic algorithm helps identify the means of triangular fuzzy numbers, which generate more accurate results. Analysis of the R2 coefficient confirms a very good fit of the proposed model. Therefore, the new non-linear normalization approach in MCDA presented in this work has great potential in adjusting decision preferences to non-monotonic criteria. Future research should continue on the accuracy of this approach in identifying non-linear decision models and apply it to other MCDA methods and real-world decision problems.

A robust TOPSIS method for decision making problems with hierarchical and non-monotonic criteria

This paper introduces an extension of a well-known Multiple Criteria Decision Aiding method, namely the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Most of the TOPSIS applications assume that preferences are monotonic for each evaluation criterion and that qualitative scales are converted into quantitative ones before the method is applied. However, both assumptions have been subject of discussion and criticism in the literature. To this solution, this paper introduces a normalization technique based on simulations that permit taking into account non-monotonic preferences as well as qualitative criteria. An additional novelty lies in the integration of the Multiple Criteria Hierarchy Process, which extends the applicability of the method to problems in which criteria are hierarchically structured. To deal with robustness concerns, the Stochastic Multicriteria Acceptability Analysis will be used in the new proposal, giving information in statistical terms on the goodness of the considered alternatives. The new method has been applied to evaluate a set of banks listed in the LSE’s FTSE350 Index.

Compromise solutions for stochastic multicriteria acceptability analysis with uncertain preferences and nonmonotonic criteria

AbstractAs one of the most popular multiple criteria decision‐making (MCDM) methods, the stochastic multicriteria acceptability analysis (SMAA) is powerful in dealing with the uncertain preference information of decision makers. However, how to find compromise solutions for MCDM problems remains an open question in the SMAA family. A compromise solution refers to an alternative that is the closest to the ideal solution. In this paper, we propose an improved SMAA method, called the SMAA with compromise solutions (SMAA‐CO), to determine compromise solutions for MCDM problems with the uncertain preferences of decision makers and conflicting criteria. The SMAA‐CO uses nonmonotonic additive value functions to depict decision makers’ ideal values for reference points. We apply the proposed method to capture multiple decision makers’ uncertain preference information. An illustrative example concerning supplier selection in the pulp and paper industry is used to show the applicability of the proposed method. Some managerial insights are given.

Preference disaggregation method for value-based multi-decision sorting problems with a real-world application in nanotechnology

We consider a problem of multi-decision sorting subject to multiple criteria. In the newly formulated decision problem, besides performances on multiple criteria, alternatives get evaluations on multiple interrelated decision attributes involving preference-ordered classes. We propose a dedicated method for dealing with such a problem, incorporating a threshold-based value-driven sorting procedure. The Decision Maker (DM) is expected to holistically evaluate a subset of reference alternatives by indicating the quality or risk level on a pre-defined scale of each decision attribute. Based on these evaluations, we construct a set of interrelated preference models, one for each decision attribute, compatible with intra- and inter-decision constraints imposed by such indirect preference information. We also formulate a new way of dealing with potentially non-monotonic criteria by discovering local monotonicity changes in different performance scale regions. The marginal value functions for criteria with unknown monotonicity are represented as a sum of two value functions assuming opposing preference directions, one non-decreasing and the other non-increasing. This permits to obtain an aggregated marginal value function with an arbitrary non-monotonic shape. The practical usefulness of the approach is demonstrated on a case study concerning risk management related to handling (i.e., production, use, manipulation, and processing) nanomaterials in different conditions. We analyze the expert judgments and discuss the inferred preference models, which can be applied to support health and safety managers in reducing the possible risk associated with the respective exposure scenario.

Preference disaggregation for multiple criteria sorting with partial monotonicity constraints: Application to exposure management of nanomaterials

We propose a novel approach to multiple criteria sorting incorporating a threshold-based value-driven procedure. The parameters deciding upon the shape of marginal value functions and separating class thresholds are inferred through preference disaggregation from the Decision Maker's incomplete assignment examples and partial requirements on the type of (non-)monotonicity for each marginal value function. These types include standard monotonic shapes, level-monotonic functions, A- and V-types combining increasing and decreasing value trends, and unknown monotonicity constraints. A representative instance of the sorting model compatible with the preference information is constructed by solving a dedicated Mixed-Integer Linear Programming problem. Its complexity is controlled by minimizing the number of changes in monotonicity between all subsequent sub-intervals of marginal value functions. The assignments derived using the constructed representative model are validated against the outcomes of robustness analysis. The proposed method is applied to a real-world problem of exposure management of engineered nanomaterials. We develop a model for predicting precaution level while handling nanomaterials in certain conditions using a respirator. The model captures interrelations between ten accounted evaluation criteria, including both monotonic and non-monotonic criteria, and the recommended class assignment. This makes it suitable for the management of exposure scenarios, which have not been directly judged by the experts.

Preference disaggregation within the regularization framework for sorting problems with multiple potentially non-monotonic criteria

We propose a new approach to preference model learning for multiple criteria sorting within the regularization framework traditionally used in the statistical learning theory. It employs an additive piecewise-linear value function as a preference model, and infers the model’s parameters from the assignment examples concerning a subset of reference alternatives. As such, our approach belongs to the family of preference disaggregation approaches. We propose a new way of measuring the complexity of the preference model. Moreover, by accounting for the trade-off between model’s complexity and fitting ability, the proposed approach avoids the problem of over-fitting and enhances the generalization ability to non-reference alternatives. In addition, it is capable of dealing with potentially non-monotonic criteria, whose marginal value functions can be inferred from the assignment examples without using integer variables. The proposed preference learning approach is formulated as a binary classification problem and addressed using support vector machine. In this way, the respective optimization problems can be solved with some computationally efficient algorithms. Moreover, the prior knowledge about the preference directions on particular criteria are incorporated to the model, and a dedicated algorithm is developed to solve the extended quadratic optimization problem. An example of university classification in China is discussed to illustrate the applicability of proposed method and extensive simulation experiments are conducted to analyze its performance under a variety of problem settings.

A progressive sorting approach for multiple criteria decision aiding in the presence of non-monotonic preferences

A new decision-aiding approach for multiple criteria sorting problems is proposed for considering the non-monotonic relationship between the preference and evaluations of the alternatives on specific criteria. The approach employs a value function as the preference model and requires the decision maker (DM) to provide assignment examples of a subset of reference alternatives as preference information. We assume that the marginal value function of a non-monotonic criterion is non-decreasing up to the criterion’s most preferred level, and then it is non-increasing. For these non-monotonic criteria, the approach starts with linearly increasing and decreasing marginal value functions but then allows such functions to deviate from the linearity and switches them to more complex ones. We develop several algorithms to help the DM resolve the inconsistency in the assignment examples and assign non-reference alternatives. The algorithms not only incorporate the DM’s evolving cognition of the preference, but also take into account the trade-offs between the capacity for satisfying incremental preference information and the complexity of the preference model. The DM is guided to evaluate the results at each iteration and then provides reactions for the subsequent iterations so that the proposed approach supports the DM to work out a satisfactory preference model. We demonstrate the applicability and validity of the proposed approach with an illustrative example and a numerical experiment.

An inexact and nonmonotone proximal method for smooth unconstrained minimization

An implementable proximal point algorithm is established for the smooth nonconvex unconstrained minimization problem. At each iteration, the algorithm minimizes approximately a general quadratic by a truncated strategy with step length control. The main contributions are: (i) a framework for updating the proximal parameter; (ii) inexact criteria for approximately solving the subproblems; (iii) a nonmonotone criterion for accepting the iterate. The global convergence analysis is presented, together with numerical results that validate and put into perspective the proposed approach.

Nonmonotone filter DQMM method for the system of nonlinear equations

In this paper, we propose a nonmonotone filter Diagonalized Quasi-Newton Multiplier (DQMM) method for solving system of nonlinear equations. The system of nonlinear equations is transformed into a constrained nonlinear programming problem which is then solved by nonmonotone filter DQMM method. A nonmonotone criterion is used to speed up the convergence progress in some ill-conditioned cases. Under reasonable conditions, we give the global convergence properties. The numerical experiments are reported to show the effectiveness of the proposed algorithm.

An extreme-distance approach to multiple criteria ranking

A distance approach based on extreme points, or predefined ideal and anti-ideal points, is proposed to improve on the TOPSIS (Technique for Order Performance [or Ordered Preference] by Similarity to Ideal Solution) method of multiple criteria ranking. Two case studies demonstrate how the analysis procedure works, and provide a basis for comparison of the proposed method to the original TOPSIS and similar methods. In applications, the new method produces results that are generally consistent with the original technique, but offers new features such as a clear interpretation of extreme points, more flexibility in setting extreme points, no normalization distortion, and the ability to handle non-monotonic criteria.

An affine scaling optimal path method with interior backtracking curvilinear technique for linear constrained optimization

This paper presents an affine scaling optimal path approach in association with nonmonotonic interior backtracking line search technique for nonlinear optimization subject to linear constraints. We shall employ the eigensystem decomposition and affine scaling mapping to form affine scaling optimal curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis is given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

Nonmonotonic Reduced Projected Hessian Method Via an Affine Scaling Interior Modified Gradient Path for Bounded-Constrained Optimization

The authors propose an affine scaling modified gradient path method in association with reduced projective Hessian and nonmonotonic interior backtracking line search techniques for solving the linear equality constrained optimization subject to bounds on variables. By employing the QR decomposition of the constraint matrix and the eigensystem decomposition of reduced projective Hessian matrix in the subproblem, the authors form affine scaling modified gradient curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The global convergence and fast local superlinear/quadratical convergence rates of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

Affine scaling interior Levenberg–Marquardt method for bound-constrained semismooth equations under local error bound conditions

We develop and analyze a new affine scaling Levenberg–Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg–Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg–Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg–Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg–Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature.

Projected Hessian algorithm with backtracking interior point technique for linear constrained optimization

In this paper, we propose a new trust-region-projected Hessian algorithm with nonmonotonic backtracking interior point technique for linear constrained optimization. By performing the QR decomposition of an affine scaling equality constraint matrix, the conducted subproblem in the algorithm is changed into the general trust-region subproblem defined by minimizing a quadratic function subject only to an ellipsoidal constraint. By using both the trust-region strategy and the line-search technique, each iteration switches to a backtracking interior point step generated by the trustregion subproblem. The global convergence and fast local convergence rates for the proposed algorithm are established under some reasonable assumptions. A nonmonotonic criterion is used to speed up the convergence in some ill-conditioned cases.

An affine scaling reduced preconditional conjugate gradient path method for linear constrained optimization

This paper presents an affine scaling reduced preconditional conjugate gradient path approach in association with nonmonotonic interior backtracking line search technique for the linear constrained optimization. Employing the affine scaling preconditional conjugate gradient to form the curvilinear path and using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis are given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions.

An affine scaling interior trust-region method for LC1 minimization subject to bounds on variables

In this literature, we extend the classical affine scaling interior trust-region algorithm for smooth bounded-constrained nonlinear programming to the nonsmooth case where the objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained LC^1 optimization subject to bounds on variables. At each iteration, the trust-region subproblem is defined by minimizing a quadratic function which is required both first- and second-order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints. The second-order derivative of the objective function is explicitly required to be Lipschitzian. Each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust-region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. Applications of the algorithm to various nonsmooth optimization problems are discussed.

Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations

We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases.

An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems

In this paper, we propose a new affine scaling trust-region algorithm in association with nonmonotonic interior backtracking line search technique for solving nonlinear equality systems subject to bounds on variables. The trust-region subproblem is defined by minimizing a squared Euclidean norm of linear model adding the augmented quadratic affine scaling term subject only to an ellipsoidal constraint. By using both trust-region strategy and interior backtracking line search technique, each iterate switches to backtracking step generated by the general trust-region subproblem and satisfies strict interior point feasibility by line search backtracking technique. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

Interior point projected reduced Hessian method with trust region strategy for nonlinear constrained optimization

A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems, a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases.