Linear Ranking Functions Research Articles

Deep Relative Attributes

Relative attribute (RA) learning aims to learn the ranking function describing the relative strength of the attribute. Most of current learning approaches learn a linear ranking function for each attribute by use of the hand-crafted visual features. Different from the existing study, in this paper, we propose a novel deep relative attributes (DRA) algorithm to learn visual features and the effective nonlinear ranking function to describe the RA of image pairs in a unified framework. Here, visual features and the ranking function are learned jointly, and they can benefit each other. The proposed DRA model is comprised of five convolutional neural layers, five fully connected layers, and a relative loss function which contains the contrastive constraint and the similar constraint corresponding to the ordered image pairs and the unordered image pairs, respectively. To train the DRA model effectively, we make use of the transferred knowledge from the large scale visual recognition on ImageNet [1] to the RA learning task. We evaluate the proposed DRA model on three widely used datasets. Extensive experimental results demonstrate that the proposed DRA model consistently and significantly outperforms the state-of-the-art RA learning methods. On the public OSR, PubFig, and Shoes datasets, compared with the previous RA learning results [2] , the average ranking accuracies have been significantly improved by about $8\%$ , $9\%$ , and $14\%$ , respectively.

Listwise Learning to Rank by Exploring Structure of Objects

Listwise learning to rank (LTR) is aimed at constructing a ranking model from listwise training data to order objects. In most existing studies, each training instance consists of a set of objects described by preference features. In a preference feature space for the objects in training, the structure of the objects is associated with the absolute preference degrees for the objects. The degrees significantly influence the ordering of the objects. Nevertheless, the structure of the training objects in their preference feature space has rarely been studied. In addition, most listwise LTR algorithms yield a single linear ranking model for all objects, but this ranking model may be insufficient to capture the underlying nonlinear ranking mechanism among all objects. This study proposes a divide-and-train method to learn a nonlinear ranking model from listwise training data. First, a rank-preserving clustering approach is used to infer the structure of objects in their preference feature space and all the objects in training data are divided into several clusters. Each cluster is assumed to correspond to a preference degree and an ordinal regression function is then learned. Second, considering that relations exist among the clusters, a multi-task listwise ranking approach is then employed to train linear ranking functions for all the clusters (or preference degrees) simultaneously. Our proposed method utilizes both the (relative) preferences among objects and the intrinsic structure of objects. Experimental results on benchmark data sets suggest that the proposed method outperforms state-of-the-art listwise LTR algorithms.

Parametric Analysis in Fuzzy Number Linear Programming Problems

This paper describes parametric analysis in fuzzy number linear programming (FNLP) when the objective function coefficients and/or the right-hand-side constants are parameterized. Using a linear ranking function, we consider the problem variations. In fact, we find a range set of the parameters for which a given basis remains optimal for the FNLP problem. If the perturbation destroys optimality and/or feasibility of the optimal basis, we use of the fuzzy primal simplex method, the fuzzy dual simplex method and/or our proposed fuzzy primal-dual simplex method to find the new optimal basis. Finally, by numerical examples we demonstrate the computational procedure.

A Fully Fuzzy DEA Approach for Cost and Revenue Efficiency Measurements in the Presence of Undesirable Outputs and Its Application to the Banking Sector in India

This paper extends the conventional cost efficiency (CE) and revenue efficiency (RE) models to fully fuzzy environments to account for real situations where input–output data and their corresponding prices are not known precisely. Owing to the importance of the presence of undesirable outputs in the production process, these are also incorporated into the production technologies of the proposed models. This paper endeavours to propose fully fuzzy CE (FFCE) and fully fuzzy RE (FFRE) models where input–output data and prices include uncertainty of fuzzy forms, in particular, of triangular membership forms. Further, the concepts of fully fuzzy linear programming problems (FFLPPs) and linear ranking functions are used to transform FFCE and FFRE models into the crisp linear programming problems (LPPs), and to evaluate fuzzy CE (FCE) and fuzzy RE (FRE) measures of the decision-making units as triangular fuzzy numbers. Moreover, the proposed models are compared with some existing approaches and are also illustrated with an application to the banking sector in India for proving their acceptability and effectiveness in real-world systems.

Synthesis of ranking functions using extremal counterexamples

We present a complete method for synthesizing lexicographic linear ranking functions (and thus proving termination), supported by inductive invariants, in the case where the transition relation of the program includes disjunctions and existentials (large block encoding of control flow). Previous work would either synthesize a ranking function at every basic block head, not just loop headers, which reduces the scope of programs that may be proved to be terminating, or expand large block transitions including tests into (exponentially many) elementary transitions, prior to computing the ranking function, resulting in a very large global constraint system. In contrast, our algorithm incrementally refines a global linear constraint system according to extremal counterexamples: only constraints that exclude spurious solutions are included. Experiments with our tool Termite show marked performance and scalability improvements compared to other systems.

Modeling and optimization of multi objective non-linear programming problem in intuitionistic fuzzy environment

In dealing with real world practical optimization problems, a decision maker usually faces a state of uncertainty as well as hesitation, due to various unpredictable factors. Sometimes it is necessary to optimize several non-linear and conflicting objectives simultaneously. To deal with the uncertain parameters which arise in such situations, intuitionistic fuzzy numbers are utilized. We formulate a multiobjective non-linear programming problem in intuitionistic fuzzy environment. We propose a linear ranking function and utilize it to convert the intuitionistic fuzzy model into a crisp model. After converting the problem into equivalent crisp problem, we propose a non-linear membership function and develop various approaches for solving it by using different operators and fuzzy programming technique. We apply our methodologies for justification to a numerical problem in manufacturing systems.

The Hardness of Finding Linear Ranking Functions for Lasso Programs

Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem is known to be in coNP when variables range over the integers and in PTIME for the rational numbers, or real numbers. Here we show that deciding whether a linear-constraint loop with a precondition, specifically with partially-specified input, has a linear ranking function is EXPSPACE-hard over the integers, and PSPACE-hard over the rationals. The precise complexity of these decision problems is yet unknown. The EXPSPACE lower bound is derived from the reachability problem for Petri nets (equivalently, Vector Addition Systems), and possibly indicates an even stronger lower bound (subject to open problems in VAS theory). The lower bound for the rationals follows from a novel simulation of Boolean programs. Lower bounds are also given for the problem of deciding if a linear ranking-function supported by a particular form of inductive invariant exists. For loops over integers, the problem is PSPACE-hard for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of natural numbers as invariants.

On solving bounded fuzzy variable linear program and its applications

In recent years various attempts have been made to study the solution of fuzzy variable linear programming problems by introducing and solving certain auxiliary problems or directly by the use of dual simplex method based on a certain linear ranking function. But these methods are not applicable for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds. In this paper, as a natural extension of the results in crisp linear programming and based on a certain linear ranking function we obtain some new results leading to a new method to overcome this shortcoming. To show the application of our method a real life problem is solved by using the proposed method. Also, one application of this approach in solving maximum flow problem with fuzzy capacity is dealt with.

Optimality Conditions for Fuzzy Number Quadratic Programming with Fuzzy Coefficients

The purpose of the present paper is to investigate optimality conditions and duality theory in fuzzy number quadratic programming (FNQP) in which the objective function is fuzzy quadratic function with fuzzy number coefficients and the constraint set is fuzzy linear functions with fuzzy number coefficients. Firstly, the equivalent quadratic programming of FNQP is presented by utilizing a linear ranking function and the dual of fuzzy number quadratic programming primal problems is introduced. Secondly, we present optimality conditions for fuzzy number quadratic programming. We then prove several duality results for fuzzy number quadratic programming problems with fuzzy coefficients.

Termination proofs for linear simple loops

Analysis of termination and other liveness properties of a program can be reduced to termination proof synthesis for simple loops, i.e., loops with only variable updates in the loop body. Among simple loops, the subset of linear simple loops (LSLs) is particularly interesting because it is common in practice and expressive in theory. Existing techniques can successfully synthesize a linear ranking function for an LSL if there exists one. However, when a terminating LSL does not have a linear ranking function, these techniques fail. In this paper, we describe an automatic method that generates proofs of (universal) termination for LSLs based on the synthesis of disjunctive ranking relations. The method repeatedly finds linear ranking functions on partitions of the state space and checks whether the transitive closure of the transition relation is included in the union of the ranking relations. Our method extends the work of Podelski and Rybalchenko (A complete method for the synthesis of linear ranking functions. In: Proceedings of the 5th international conference on VMCAI, Jan 2004, Venice, Italy, pp 239---251, 2004). We have implemented a prototype of the method and have shown experimental evidence of the effectiveness of our method.

Ranking function synthesis for bit-vector relations

Ranking function synthesis is a key component of modern termination provers for imperative programs. While it is well-known how to generate linear ranking functions for relations over (mathematical) integers or rationals, efficient synthesis of ranking functions for machine-level integers (bit-vectors) is an open problem. This is particularly relevant for the verification of low-level code. We propose several novel algorithms to generate ranking functions for relations over machine integers: a complete method based on a reduction to Presburger arithmetic, and a template-matching approach for predefined classes of ranking functions based on reduction to SAT- and QBF-solving. The utility of our algorithms is demonstrated on examples drawn from Windows device drivers.

On the linear ranking problem for integer linear-constraint loops

In this paper we study the complexity of the Linear Ranking problem: given a loop, described by linear constraints over a finite set of integer variables, is there a linear ranking function for this loop? While existence of such a function implies termination, this problem is not equivalent to termination. When the variables range over the rationals or reals, the Linear Ranking problem is known to be PTIME decidable. However, when they range over the integers, whether for single-path or multipath loops, the complexity of the Linear Ranking problem has not yet been determined. We show that it is coNP-complete. However, we point out some special cases of importance of PTIME complexity. We also present complete algorithms for synthesizing linear ranking functions, both for the general case and the special PTIME cases.

Interior Point Method for Solving Fuzzy Number Linear Programming Problems Using Linear Ranking Function

Recently, various methods have been developed for solving linear programming problems with fuzzy number, such as simplex method and dual simplex method. But their computational complexities are exponential, which is not satisfactory for solving large-scale fuzzy linear programming problems, especially in the engineering field. A new method which can solve large-scale fuzzy number linear programming problems is presented in this paper, which is named a revised interior point method. Its idea is similar to that of interior point method used for solving linear programming problems in crisp environment before, but its feasible direction and step size are chosen by using trapezoidal fuzzy numbers, linear ranking function, fuzzy vector, and their operations, and its end condition is involved in linear ranking function. Their correctness and rationality are proved. Moreover, choice of the initial interior point and some factors influencing the results of this method are also discussed and analyzed. The result of algorithm analysis and example study that shows proper safety factor parameter, accuracy parameter, and initial interior point of this method may reduce iterations and they can be selected easily according to the actual needs. Finally, the method proposed in this paper is an alternative method for solving fuzzy number linear programming problems.

A new look at the automatic synthesis of linear ranking functions

The classical technique for proving termination of a generic sequential computer program involves the synthesis of a ranking function for each loop of the program. Linear ranking functions are particularly interesting because many terminating loops admit one and algorithms exist to automatically synthesize it. In this paper we present two such algorithms: one based on work dated 1991 by Sohn and Van Gelder; the other, due to Podelski and Rybalchenko, dated 2004. Remarkably, while the two algorithms will synthesize a linear ranking function under exactly the same set of conditions, the former is mostly unknown to the community of termination analysis and its general applicability has never been put forward before the present paper. In this paper we thoroughly justify both algorithms, we prove their correctness, we compare their worst-case complexity and experimentally evaluate their efficiency, and we present an open-source implementation of them that will make it very easy to include termination-analysis capabilities in automatic program verifiers.

A primal-dual method for solving linear programming problems with fuzzy cost coefficients based on linear ranking functions and its applications

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

A Novel Algorithm For Finding Equilibrium Strategy In Two Person Zero Sum Game With Fuzzy Strategy Sets And Payoffs

This paper investigates a two person zero sum matrix game in which the payoffs and strategy are characterized as random fuzzy variables. Using the operations of triangular fuzzy numbers, the fuzzy payoffs for all synthetic outcomes are calculated. Based on random fuzzy expected value operator, a random fuzzy expected minimax equilibrium strategy to the game is defined. After that, based on the constraints, the feasible strategy string sets of the players for multi conflict situations are constructed. Then an iterative algorithm based on random fuzzy simulation is designed to seek the minimax equilibrium strategy. Using a linear ranking function, the aggregation model can be solved by transforming it into a crisp bimatrix game. Then, the fuzzy synthetic aggregation model is established and solved by transforming it into a crisp bimatrix game. Finally, a military example is provided to illustrate the practicality and effectively of the model.

Sensitivity analysis in fuzzy number linear programming problems

In this paper, we generalize the concept of sensitivity analysis in fuzzy number linear programming (FLNP) problems by applying fuzzy simplex algorithms and using the general linear ranking functions on fuzzy numbers. The purpose of sensitivity analysis is to determine changes in the optimal solution of FNLP problem resulting from changes in the data. If the change affects the optimality of the basis, we perform primal pivots to achieve optimality by use of the fuzzy primal simplex method. Whenever the change destroys the feasibility of the optimal basis, we perform dual pivots to achieve feasibility by use of the fuzzy dual simplex method.

FUZZY NONLINEAR OPTIMIZATION

In this paper, we define a fuzzy version of the quadratic optimization problem and then give a method to solve these problems by using linear ranking functions. The method will be discussed in details.

Size-change termination with difference constraints

This article considers an algorithmic problem related to the termination analysis of programs. More specifically, we are given bounds on differences in sizes of data values before and after every transition in the program's control-flow graph. Our goal is to infer program termination via the following reasoning (“the size-change principle”): if in any infinite (hypothetic) execution of the program, some size must descend unboundedly, the program must always terminate, since infinite descent of a natural number is impossible. The problem of inferring termination from such abstract information is not the halting problem for programs and may well be decidable. If this is the case, the decision algorithm forms a “back end” of a termination verifier, and it is interesting to find out the computational complexity of the problem. A restriction of the problem described above, which only uses monotonicity information (but not difference bounds), is already known to be decidable. We prove that the unrestricted problem is undecidable, which gives a theoretical argument for studying restricted cases. We consider a case where the termination proof is allowed to make use of at most one bound per target variable in each transition. For this special case, which we claim is practically significant, we give (for the first time) an algorithm and show that the problem is in PSPACE, in fact that it is PSPACE-complete. The algorithm is based on combinatorial arguments and results from the theory of integer programming not previously used for similar problems. The algorithm has interesting connections to other work in termination, in particular to methods for generating linear ranking functions or invariants.

Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables

Linear programming problems with trapezoidal fuzzy variables (FVLP) have recently attracted some interest. Some methods have been developed for solving these problems by introducing and solving certain auxiliary problems. Here, we apply a linear ranking function to order trapezoidal fuzzy numbers. Then, we establish the dual problem of the linear programming problem with trapezoidal fuzzy variables and hence deduce some duality results. In particular, we prove that the auxiliary problem is indeed the dual of the FVLP problem. Having established the dual problem, the results will then follow as natural extensions of duality results for linear programming problems with crisp data. Finally, using the results, we develop a new dual algorithm for solving the FVLP problem directly, making use of the primal simplex tableau. This algorithm will be useful for sensitivity (or post optimality) analysis when using primal simplex tableaus.