/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

/pdf/engineering-and-economic-applications-of-complementarity-1nz165yv0c.pdf

Engineering and Economic Applications of Complementarity Problems

An algorithm is proposed which generates a nonlinear kernel-based separating surface that requires as little as 1% of a large dataset for its explicit evaluation. To generate this nonlinear surface, the entire dataset is used as a constraint in an optimization problem with very few variables corresponding to the 1% of the data kept. The remainder of the data can be thrown away after solving the optimization problem. This is achieved by making use of a rectangular m×m kernel K(A, Ā) that greatly reduces the size of the quadratic program to be solved and simplifies the characterization of the nonlinear separating surface. Here, the m rows of A represent the original m data points while the m rows of Ā represent a greatly reduced m data points. Computational results indicate that test set correctness for the reduced support vector machine (RSVM), with a nonlinear separating surface that depends on a small randomly selected portion of the dataset, is better than that of a conventional support vector machine (SVM) with a nonlinear surface that explicitly depends on the entire dataset, and much better than a conventional SVM using a small random sample of the data. Computational times, as well as memory usage, are much smaller for RSVM than that of a conventional SVM using the entire dataset.

/pdf/rsvm-reduced-support-vector-machines-30r80bxpp5.pdf

RSVM: Reduced Support Vector Machines

Multicriteria classification and sorting methods: A literature review

Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform.

/pdf/a-direct-method-for-trajectory-optimization-of-rigid-bodies-d1xf28iljh.pdf

A direct method for trajectory optimization of rigid bodies through contact

We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)+=max{0, x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter α. Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter α, generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].

/pdf/a-class-of-smoothing-functions-for-nonlinear-and-mixed-lpks1dp0xy.pdf

A class of smoothing functions for nonlinear and mixed complementarity problems

A smooth approximationp (x, α) to the plus function max{x, 0} is obtained by integrating the sigmoid function 1/(1 + e−αx
), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy forα sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finiteα. Speedup over MINOS 5.4 was as high as 1142 times for linear inequalities of size 2000 × 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCPs with as many as 10 000 variables, the proposed approach was as much as 63 times faster than Lemke's method.

/pdf/smoothing-methods-for-convex-inequalities-and-linear-1vymcon2be.pdf

Smoothing methods for convex inequalities and linear complementarity problems

Given two finite point setsA andB in then-dimensional real spaceR

 n
 , we consider the NP-complete problem of minimizing the number of misclassified points by a plane attempting to divideR

 n
 into two halfspaces such that each open halfspace contains points mostly ofA orB. This problem is equivalent to determining a plane {x | x

 T
 
w=γ} that maximizes the number of pointsx ∈A satisfying inx

 T
 
w>γ, plus the number of pointsx ∈B satisfyingx

 T
 
w<γ. A simple but fast algorithm is proposed that alternates between (i) minimizing the number of misclassified points by translation of the separating plane, and (ii) a rotation of the plane so that it minimizes a weighted average sum of the distances of the misclassified points to the separating plane. Existence of a global solution to an underlying hybrid minimization problem is established. Computational comparison with a parametric approach to solve the NP-complete problem indicates that our approach is considerably faster and appears to generalize better as determined by tenfold cross-validation.

/pdf/hybrid-misclassification-minimization-23dvtf7wj9.pdf

Chunhui Chen

Papers

A class of smoothing functions for nonlinear and mixed complementarity problems

Smoothing methods for convex inequalities and linear complementarity problems

Hybrid misclassification minimization