Piecewise Linear Programming Research Articles

굴곡진 도로에서의 구간 선형 모델을 이용한 차선 검출

본 논문에서는 굴곡진 도로를 구간 선형 모델로 근사화한 차선 검출 알고리즘을 제안한다. 기존의 차선 검출 알고리즘들은 지표면이 평면이라는 가정을 이용하기 때문에, 도로면이 굴곡진 실제 도로에서는 강건한 차선 검출이 어렵다. 제안하는 방법에서는 이 문제를 전체 차로를 구간으로 분할하고, 각 구간 내에서 차로를 가장 잘 근사하는 평면 차로를 구함으로써 해결한다. 이를 위해 각 구간 내에서 다양한 각도와 위치를 가지는 평면 형태의 구간 차로 후보들을 생성하였다. 구간 차로 후보들의 연결 조합 중 실제 차로에 가장 가까운 조합을 다이나믹프로그래밍을 이용하여 찾음으로써 굴곡진 차로를 근사한다. 평면 도로 뿐 아니라, 상하, 좌우의 굴곡이 있는 도로 영상으로 구성된 데이터세트에 대하여 제안하는 방법의 차선 검출 성능을 검증하였다. 평면 도로를 가정한 기존의 방법들이 80%에서 90% 초반의 검출률을 보이는 반면, 제안하는 방법은 90% 후반의 검출률을 보임을 통해 굴곡진 도로에서의 차선 검출의 강건성을 입증하였다. This paper proposes a robust lane detection algorithm for non-flat roads by combining a piecewise linear model and dynamic programming. Compared with other lane models, the piecewise linear model can represent 3D shapes of roads from the scenes acquired by monocular camera since it can form a curved surface through a set of planar road. To represent the real road, the planar roads are created by various angles and positions at each section. And dynamic programming determines an optimal combination of planar roads based on lane properties. Experiment results demonstrate the robustness of proposed algorithm against non-flat road, curved road, and camera vibration.

An algorithm for the topology optimization of geometrically nonlinear structures

SUMMARYMost papers on topology optimization consider that there is a linear relation between the strains and displacements of the structure, implicitly assuming that the displacements of the structure are small. However, when the external loads applied to the structure are large, the displacements also become large, so it is necessary to suppose that there is a nonlinear relation between strains and displacements. In this case, we say that the structure is geometrically nonlinear. In practice, this means that the linear system that needs to be solved each time the objective function of the problem is evaluated is replaced by an ill‐conditioned nonlinear system of equations. Moreover, the stiffness matrix and the derivatives of the problem also become harder to compute.The objective of this work is to solve topology optimization problems under large displacements through a new optimization algorithm, named sequential piecewise linear programming. This method relies on the solution of convex piecewise linear programming subproblems that include second order information about the objective function. To speed up the algorithm, these subproblems are converted into linear programming ones. The new algorithm is not only globally convergent to stationary points but our numerical experiments also show that it is efficient and robust. Copyright © 2014 John Wiley & Sons, Ltd.

Value-based data envelopment analysis: a piece-wise linear programming approach

Incorporating value judgments in data envelopment analysis (DEA) is a broad methodological framework that facilitates driving the efficiency assessments in line with individual preferences. Value-based DEA is a recent development that resorts to value assessment protocols from multiple criteria decision analysis (MCDA) to transform the original input/output data to a value scale. In this context, we introduce in this paper a piece-wise linear programming approach to value-based DEA, by employing a data transformation-variable alteration technique and assurance region constraints. We outline our approach by revisiting a previous efficiency assessment study drawn from the literature.

Loss-Aversion with Kinked Linear Utility Functions

Prospect theory postulates that the utility function is characterized by a kink (a point of non-differentiability) that distinguishes gains from losses. In this paper we present an algorithm that efficiently solves the linear version of the kinked-utility problem. First, we transform the non-differentiable kinked linear-utility problem into a higher dimensional, differentiable, linear program. Second, we introduce an efficient algorithm that solves the higher dimensional linear program in a smaller dimensional space. Third, we employ a numerical example to show that solving the problem with our algorithm is 15 times faster than solving the higher dimensional linear program using the interior point method of Mosek. Finally, we provide a direct link between the piece-wise linear programming problem and conditional value-at-risk, a downside risk measure.

Dynamic programming in parallel boundary detection with application to ultrasound intima-media segmentation

Segmentation of carotid artery intima-media in longitudinal ultrasound images for measuring its thickness to predict cardiovascular diseases can be simplified as detecting two nearly parallel boundaries within a certain distance range, when plaque with irregular shapes is not considered. In this paper, we improve the implementation of two dynamic programming (DP) based approaches to parallel boundary detection, dual dynamic programming (DDP) and piecewise linear dual dynamic programming (PL-DDP). Then, a novel DP based approach, dual line detection (DLD), which translates the original 2-D curve position to a 4-D parameter space representing two line segments in a local image segment, is proposed to solve the problem while maintaining efficiency and rotation invariance. To apply the DLD to ultrasound intima-media segmentation, it is imbedded in a framework that employs an edge map obtained from multiplication of the responses of two edge detectors with different scales and a coupled snake model that simultaneously deforms the two contours for maintaining parallelism. The experimental results on synthetic images and carotid arteries of clinical ultrasound images indicate improved performance of the proposed DLD compared to DDP and PL-DDP, with respect to accuracy and efficiency.

Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs

In this paper, we establish the equivalence between the half-space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions. As consequences, we prove that the corresponding marginal function is differentiable and the solution map admits a differentiable selection.

Piecewise Linear Multicriteria Programs: The Continuous Case and Its Discontinuous Generalization

In this paper we study piecewise linear multicriteria programs, that is, multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint. We obtain an algebraic representation of a semi-closed polyhedron and apply it to show that the image of a semi-closed polyhedron under a continuous linear function is always one semi-closed polyhedron. We establish that the (weak) Pareto solution/point set of a piecewise linear multicriteria program is the union of finitely many semi-closed polyhedra. We propose an algorithm for finding the Pareto point set of a continuous piecewise linear bi-criteria program and generalize it to the discontinuous case. We apply our algorithm to solve the discontinuous bi-criteria portfolio selection problem with an l∞ risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy.

In This Issue

In This Issue

Exact Penalty and Optimality Condition for Nonseparable Continuous Piecewise Linear Programming

Utilizing compact representations for continuous piecewise linear functions, this paper discusses some theoretical properties for nonseparable continuous piecewise linear programming. The existence of exact penalty for continuous piecewise linear programming is proved, which allows us to concentrate on unconstrained problems. For unconstrained problems, we give a sufficient and necessary local optimality condition, which is based on a model with universal representation capability and hence applicable to arbitrary continuous piecewise linear programming. From the gained optimality condition, an algorithm is proposed and evaluated by numerical experiments, where the theoretical properties are illustrated as well.

A new risk criterion in fuzzy environment and its application

Since the observed values of security returns in real-world problems are sometimes imprecise or vague, an increasing effort in research is devoted to study the properties of risk measures in fuzzy portfolio optimization problems. In this paper, a new risk measure is suggested to gauge the risk resulted from fuzzy uncertainty. For this purpose, the absolute deviation and absolute semi-deviation are first defined for fuzzy variable by nonlinear fuzzy integrals. To compute effectively the absolute semi-deviations of single fuzzy variable as well as its functions, this paper discusses the methods of computing the absolute semi-deviation by classical Lebesgue–Stieltjes (L–S) integral. After that, several useful absolute deviation and absolute semi-deviation formulas are established for common triangular, trapezoidal and normal fuzzy variables. Applying the absolute semi-deviation as a new risk measure in portfolio optimization, three classes of fuzzy portfolio optimization models are developed by combining the absolute semi-deviation with expected value operator and credibility measure. Based on the analytical representation of absolute semi-deviations, the established fuzzy portfolio selection models can be turned into their equivalent piecewise linear or fractional programming problems. Since the absolute semi-deviation is a piecewise fractional function and pseudo-convex on the feasible subregions of deterministic programming models, we take advantage of the structural characteristics to design a domain decomposition method to separate a deterministic programming problem into three convex subproblems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness of the solution method.

Global optimization method for mixed transportation network design problem: A mixed-integer linear programming approach

This paper proposes a global optimization algorithm for solving a mixed (continuous/discrete) transportation network design problem (MNDP), which is generally expressed as a mathematical programming with equilibrium constraint (MPEC). The upper level of the MNDP aims to optimize the network performance via both expansion of existing links and addition of new candidate links, whereas the lower level is a traditional Wardrop user equilibrium (UE) problem. In this paper, we first formulate the UE condition as a variational inequality (VI) problem, which is defined from a finite number of extreme points of a link-flow feasible region. The MNDP is approximated as a piecewise-linear programming (P-LP) problem, which is then transformed into a mixed-integer linear programming (MILP) problem. A global optimization algorithm based on a cutting constraint method is developed for solving the MILP problem. Numerical examples are given to demonstrate the efficiency of the proposed method and to compare the results with alternative algorithms reported in the literature.

Postoptimal analysis in the coefficients matrix piecewise linear fractional programming problems with non-degenerate optimal solution

In this paper, we discuss how changes in the coefficients matrix of piecewise linear fractional programming problems affect the non-degenerate optimal solution. We consider separate cases when changes occur in the coefficients of the basic and non-basic variables and derive bounds for each perturbation, while the optimal solution is invariant. We explain that this analysis is a generalization of the sensitivity analysis for \(LP\), \(LFP\) and \(PLP\). Finally, the results are described by some numerical examples.

Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming

In this paper, we study multi-parametric sensitivity analysis for programming problems with the piecewise linear fractional objective function using the concept of maximum volume in the tolerance region. We construct critical regions (the set of parameters values which the coefficients matrix of the problem (PLFP) may vary while still retaining the same optimal basis B.) for simultaneous and independent perturbations of one row or one column of the constraint matrix in the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as 'focal' and 'non-focal'. Non-focal parameters can be deleted from the analysis, because of their low sensitivity in practice. Theoretical results are illustrated with the help of a numerical example.

A simplex algorithm for network flow problems with piecewise linear fractional objective function

The simplex algorithm for network flow problems has been proposed for linear, piecewise linear, and linear fractional objective functions. Piecewise linear fractional network flow problems can be transformed into equivalent linear or linear fractional programs, but the resulting problems do not possess pure network flow structure and increase the number of variables significantly. We develop a simplex algorithm to solve network flow problems with piecewise linear fractional objective function, unifying and generalizing the network simplex algorithm, network simplex algorithm for piece-wise linear programs, and network simplex algorithm for linear fractional programs. Computational results using the proposed algorithm are also discussed.

Sensitivity analysis in piecewise linear fractional programming problem with non-degenerate optimal solution

In this paper, we study how changes in the coefficients of objective function and the right-hand-side vector of constraints of the piecewise linear fractional programming problems affect the non-degenerate optimal solution. We consider separate cases when changes occur in different parts of the problem and derive bounds for each perturbation, while the optimal solution is invariant. We explain that this analysis is a generalization of the sensitivity analysis for \(LP\), \(LFP\) and \(PLP\). Finally, the results are described by some numerical examples.

Inf–sup control of discontinuous piecewise affine systems

AbstractThis paper considers the worst‐case optimal control of discontinuous piecewise affine (PWA) systems, which are subjected to constraints and disturbances. We seek to pre‐compute, via dynamic programming, an explicit control law for these systems when a PWA cost function is utilized. One difficulty with this problem class is that, even for initial states for which the value function of the optimal control problem is finite, there might not exist a control law that attains the infimum. Hence, we propose a method that is guaranteed to obtain a sub‐optimal solution, and where the degree of sub‐optimality can be specified a priori. This is achieved by approximating the underlying sub‐problems with a parametric piecewise linear program. Copyright © 2008 John Wiley & Sons, Ltd.

Multi-parametric sensitivity analysis in piecewise linear fractional programming

In this paper, we study multi-parametric sensitivity analysis forprogramming problems with the piecewise linear fractional objective function in the tolerance region. We construct critical regions for simultaneous and independent perturbations in the objective function coefficients and in the right-hand-side vector of the given problem. Necessary and sufficient conditions are derived to classify perturbation parameters as 'focal' and 'non-focal'. Non-focal parameters can be deleted from the analysis, because of their low sensitivity in practice. Theoretical results are illustrated with the help of a numerical example.

Neural Network Based Models for Efficiency Frontier Analysis: An Application to East Asian Economies’ Growth Decomposition

There has been a long tradition in business and economics to use frontier analysis to assess a production unit's performance. The first attempt utilized the data envelopment analysis (DEA) which is based on a piecewise linear and mathematical programming approach, whilst the other employed the parametric approach to estimate the stochastic frontier function. Both approaches have their advantages as well as limitations. This paper sets out to use an alternative approach, i.e. artificial neural networks (ANNs) for measuring efficiency and productivity growth for seven East Asian economies at manufacturing level, for the period 1963 to 1998, and the relevant comparisons are carried out between DEA and ANN, and stochastic frontier analysis (SFA) and ANN in order to test the ability of ANNs to assess the performance of production units. The results suggest that ANNs are a promising alternative to traditional approaches, to approximate production functions more accurately and measure efficiency and productivity under non-linear contexts, with minimum assumptions.

A simplex algorithm for piecewise-linear fractional programming problems

Generalizations of the well-known simplex method for linear programming are available to solve the piecewise linear programming problem and the linear fractional programming problem. In this paper we consider a further generalization of the simplex method to solve piecewise linear fractional programming problems unifying the simplex method for linear programs, piecewise linear programs, and the linear fractional programs. Computational results are presented to obtain further insights into the behavior of the algorithm on random test problems.

An optimal piecewise-linear program for the U-line balancing problem with stochastic task times

The utilization of U-shaped layouts in place of the traditional straight-line configuration has become increasingly popular, with reported benefits of substantial improvements in productivity and quality. This paper examines the U-line balancing problem with stochastic task times. A chance-constrained, piecewise-linear, integer program is formulated to find the optimal solution. Various approaches used to identify a tight lower bound are also presented, as are operable extensions to the basic model. Computational results show that the proposed method is able to solve practical-sized problems.