Complementarity Function Research Articles

A Neural Network Approach for Solving Weighted Nonlinear Complementarity Problems

In this paper, we consider the weighted nonlinear complementarity problem which is the extension of the general complementarity problems and contains a wide class of optimization problems. Based on the weighted complementarity function, we reformulate the weighted nonlinear complementarity problem as an unconstrained minimization problem and present the steepest descent-based neural network to solve it. Under mild conditions, we derive some results related to the asymptotic stability of the neural network. Numerical experiments indicate that the proposed algorithm is quite effective.

A least-change secant algorithm for solving generalized complementarity problem

AbstractIn this paper, we propose a least-change secant algorithm to solve the generalized complementarity problem indirectly trough its reformulation as a nonsmooth system of nonlinear equations using a one-parametric family of complementarity functions. We present local and superlinear convergence results of new algorithm and analyze its numerical performance.

New smooth weighted complementarity functions and a cubically convergent method for wLCP

New smooth weighted complementarity functions and a cubically convergent method for wLCP

One-parametric class of smoothing functions for (WSOCCP) and its Jacobian consistency

In this paper, we are interested in the study of weighted second-order cone complementarity problem (WSOCCP) and the smoothing function associated with it. A weighted second-order cone complementarity problem is to find a pair of vectors ( x , y ) that belong to the intersection of a manifold and a second-order cone such that the product of the vectors in a certain algebra, x ∘ y equals a given weight vector w. It is obvious to see that if w is zero, we get a second-order cone complementarity problem (SOCCP). Thus, (WSOCCP) provides the essential framework for a broader class of problems arising from real applications (science and engineering) more conveniently. This paper demonstrates the Jacobian consistency of the one-parametric class of smoothing functions. Furthermore, we provide an estimation of the distance between the subgradient of the one-parametric class of the weighted SOC complementarity functions and the gradient of its associated smoothing function. This estimation facilitates to adjust an appropriate parameter in smoothing methods.

Discrete-time Euler-smoothing methods for time-varying convex constrained optimization

In this paper, we propose two kinds of prediction–correction methods, discrete-time Euler-smoothing methods, for solving the general time-varying convex constrained optimization problem. By using the complementarity function and smoothing technique, the Karush–Kuhn–Tucker (KKT) trajectory of the problem can be reformulated as a system of smooth equations and considering the dynamics of the system of smooth equations in a discrete constant time-sampling periods scheme. The proposed methods mainly consist of the prediction step derived by Euler approximation and the correction step generated by one or multiple Newton iterations. Under some reasonable conditions, the approximate solution trajectories produced by the methods achieve the asymptotic error bound behaving as O(h4) with respect to optimal trajectory both in the primal and dual space as the smoothing parameter approaches to zero, where h is the sampling period. Finally, the experimental results show that our methods are efficient, and the steady-state tracking errors are much smaller than the other prediction–correction methods under the same conditions.

A nonsmooth Newton method for solving the generalized complementarity problem

In this paper, we present a new nonsmooth Newton-type algorithm for solving the generalized complementarity problem based on its reformulation as a system of nonlinear equations using a one-parametric family of complementarity functions. We demonstrate, under suitable hypotheses, that this algorithm converges locally and q-quadratically. In addition, we show numerical experiments that allow us to see the good performance of the proposed algorithm.

The Alternating Direction Search Pattern Method for Solving Constrained Nonlinear Optimization Problems

We adopt the alternating direction search pattern method to solve the equality and inequality constrained nonlinear optimization problems. Firstly, a new augmented Lagrangian function with a nonlinear complementarity function is proposed to transform the original constrained problem into a new unconstrained problem. Under appropriate conditions, it has been proven that there is a 1-1 correspondence between the local and global optimal solutions of the new unconstrained problem and the original constrained problem. In this way, the optimal solution of the original problem can be obtained by solving the new unconstrained optimization problem. Furthermore, based on the characteristics of the new problem, the alternating direction pattern search method was designed and its convergence was demonstrated. Numerical experiments were implemented to illustrate the availability of the new augmented Lagrangian function and the algorithm.

A Generalized Projetion Gradient Algorithm for Mathematical Programs with Complementary Constraints

Against the shortcomings that many existing algorithms for solving the standard smoothing nonlinear programming would fail if they were used directly to solve the mathematical programs with complementary constraints( MPCC). By using a complementarity function and the idea of smoothing approximation method, the MPCC problem was transformed into a smoothing nonlinear programming. Combined with the supermemory gradient idea, a generalized projection gradient algorithm is proposed and its global convergence is obtained.

Can the jigsaw puzzle model of protein folding re-assemble a hydrophobic core?

According to the "jigsaw puzzle" model of protein folding, the isomorphism between sequence and structure is substantially determined by the specific geometry of side-chain interactions, within the protein interior. In this work, we have attempted to predict the hydrophobic core of cyclophilin (LdCyp) from Leishmania donovani, utilizing a surface complementarity function, which selects for high goodness of fit between hydrophobic side-chain surfaces, rather in the manner of assembling a three-dimensional jigsaw puzzle. The computational core prediction method implemented here has been tried on two distinct scenarios, on the LdCyp polypeptide chain with native non-core residues and all core residues initially set to alanine, on a poly-glycine polypeptide chain. Molecular dynamics simulations appeared to indicate partial destabilization of the two designed sequences. However, experimental characterization of the designed sequences by circular dichroism (CD) spectroscopy and denaturant (GdmCl) induced unfolding, demonstrated disordered proteins. Stepwise reconstruction of the designed cores by cumulative sequential mutations identified the specific mutation (M122L) as primarily responsible for fold collapse and all design objectives were achieved upon rectifying this mutation. In summary, the study demonstrates regions of the core to contain highly specific (jigsaw puzzle-like) interactions sensitive to any perturbations and a predictive algorithm to identify such regions. A mutation within the core has been identified which exercises an inordinate influence on the global fold, reminiscent of metamorphic proteins. In addition, the computational procedure could predict substantial regions of the core (given main-chain coordinates) without any reference to non-core residues.

A smooth Levenberg-Marquardt method without nonsingularity condition for wLCP

<abstract><p>In this paper we consider the weighted Linear Complementarity Problem (wLCP). By using a smooth weighted complementarity function, we reformulate the wLCP as a smooth nonlinear equation and propose a Levenberg-Marquardt method to solve it. The proposed method differentiates itself from the current Levenberg-Marquardt type methods by adopting a simple derivative-free line search technique. It is shown that the proposed method is well-defined and it is globally convergent without requiring wLCP to be monotone. Moreover, the method has local sub-quadratic convergence rate under the local error bound condition which is weaker than the nonsingularity condition. Some numerical results are reported.</p></abstract>

Novel Nonlinear Complementarity Function Approach for Mechanical Analysis of Tensegrity Structures

Cable slacking may be frequently encountered in mechanical analysis of tensegrity structures. Traditional nonlinear finite element analysis in handling this issue may lead to numerical difficulty due to the dramatic changes of the stiffness matrix induced by cables switching between taut and slack states. To deal with the nonsmooth nature induced by cable slacking, we propose in this paper a novel nonlinear complementarity function-based framework for mechanical analysis of tensegrity structures. The core idea is that the constitutive relation of cables is reformulated as a set of equality equations. This is done by first introducing a parametric variable and expressing the constitutive relation with a complementarity condition, and then using the modified Fischer–Burmeister complementarity function. By considering the introduced parametric variables as additional degrees of freedom in the finite element model, it becomes possible to get rid of the nonsmooth nature, as well as to solve a continuous, smooth, differentiable problem by using the classical Newton–Raphson method. The proposed framework is first developed for static analysis and then extended to dynamic analysis, both under large displacement. The closed-form expressions of the Jacobian matrix are analytically derived. Multiple examples are presented to verify the robustness and effectiveness of the proposed framework.

Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

A Double Nonmonotone Quasi-Newton Method for Nonlinear Complementarity Problem Based on Piecewise NCP Functions

In this paper, a double nonmonotone quasi-Newton method is proposed for the nonlinear complementarity problem. By using 3-1 piecewise and 4-1 piecewise nonlinear complementarity functions, the nonlinear complementarity problem is reformulated into a smooth equation. By a double nonmonotone line search, a smooth Broyden-like algorithm is proposed, where a single solution of a smooth equation at each iteration is required with the reduction in the scale of the calculation. Under suitable conditions, the global convergence of the algorithm is proved, and numerical results with some practical applications are given to show the efficiency of the algorithm.

Integration of time-series meta-omics data reveals how microbial ecosystems respond to disturbance

The development of reliable, mixed-culture biotechnological processes hinges on understanding how microbial ecosystems respond to disturbances. Here we reveal extensive phenotypic plasticity and niche complementarity in oleaginous microbial populations from a biological wastewater treatment plant. We perform meta-omics analyses (metagenomics, metatranscriptomics, metaproteomics and metabolomics) on in situ samples over 14 months at weekly intervals. Based on 1,364 de novo metagenome-assembled genomes, we uncover four distinct fundamental niche types. Throughout the time-series, we observe a major, transient shift in community structure, coinciding with substrate availability changes. Functional omics data reveals extensive variation in gene expression and substrate usage amongst community members. Ex situ bioreactor experiments confirm that responses occur within five hours of a pulse disturbance, demonstrating rapid adaptation by specific populations. Our results show that community resistance and resilience are a function of phenotypic plasticity and niche complementarity, and set the foundation for future ecological engineering efforts.

A control problem approach to Coulomb’s friction

In this work we present a formulation of Coulomb’s friction in a fractured elastic body as a PDE control problem where the observed quantity is the tangential stress across an internal interface, while the control parameter is the slip i.e. the displacement jump across the interface. The cost function aims at minimizing the norm of a non-linear and not everywhere differentiable complementarity function, written in terms of the tangential stress and the slip. The interesting point of this method is that gives rise to an iterative procedure where at each iteration we solve a problem with given slip at the interface, without resorting to the use of Lagrange multipliers. We carry out a formal derivation of the method, with some preliminary results, and a numerical experiment to verify the efficacy of the technique.

A New Algorithm to Solve the Generalized Nash Equilibrium Problem

We try a new algorithm to solve the generalized Nash equilibrium problem (GNEP) in the paper. First, the GNEP is turned into the nonlinear complementarity problem by using the Karush–Kuhn–Tucker (KKT) condition. Then, the nonlinear complementarity problem is converted into the nonlinear equation problem by using the complementarity function method. For the nonlinear equation equilibrium problem, we design a coevolutionary immune quantum particle swarm optimization algorithm (CIQPSO) by involving the immune memory function and the antibody density inhibition mechanism into the quantum particle swarm optimization algorithm. Therefore, this algorithm has not only the properties of the immune particle swarm optimization algorithm, but also improves the abilities of iterative optimization and convergence speed. With the probability density selection and quantum uncertainty principle, the convergence of the CIQPSO algorithm is analyzed. Finally, some numerical experiment results indicate that the CIQPSO algorithm is superior to the immune particle swarm algorithm, the Newton method for normalized equilibrium, or the quasivariational inequalities penalty method. Furthermore, this algorithm also has faster convergence and better off-line performance.

Sliding cable modeling: A nonlinear complementarity function based framework

Sliding cables are widespread in engineering applications. The friction between the cable and the sliding components results in the existence of diversified motion states of the contact points, which leads to many difficult problems for traditional analysis methods. This paper proposes an effective and robust nonlinear complementarity function based framework for static and dynamic analyses of sliding cable problems. The core idea is that the involved sliding criterion is expressed by complementarity relationships which can be further reformulated as a set of equation by using the modified Fischer-Burmeister complementarity function. By considering the instrumental sliding lengths as additional degrees of freedom in the finite element model, the sliding nodes searching and the geometric configuration searching in the analysis can be handed in a unified framework. The solution can be obtained by using classical Newton-Raphson scheme with a closed-form expression of tangent matrix. Cumbersome trial and error iterations for sliding node searching in traditional analysis methods are rigorously avoided, and always the same equation is used regardless of whether a particular contact point would be sticking or sliding. This framework is developed mainly for implicit static and dynamic analyses, but can also be integrated into explicit analyses. Some typical examples are presented to illustrate the robustness and versatility of the proposed approach. The result highlight its ability to capture the complex and diverse motion states of the contact points in a rather direct way.

A local Jacobian smoothing method for solving Nonlinear Complementarity Problems

In this paper, we present a smoothing of a family of nonlinear complementarity functions and use its properties in combination with the smooth Jacobian strategy to present a new generalized Newton-type algorithm to solve a nonsmooth system of equations equivalent to the Nonlinear Complementarity Problem. In addition, we prove that the algorithm converges locally and q-quadratically, and analyze its numerical performance.

On construction of new NCP functions

We report a new method to construct complementarity functions for the nonlinear complementarity problem (NCP). Basic properties related to growth behavior, convexity and semismoothness of the newly discovered NCP functions are proved. We also present some variants, generalizations and other transformations of these NCP functions. Finally, we propose some interesting research directions that can be explored in the NCP research.

Returning to the Gender Frontier

Returning to the Gender Frontier Susanah Shaw Romney (bio) Tai S. Edwards. Osage Women and Empire: Gender and Power. Lawrence: University Press of Kansas, 2018. vii + 232 pp.; ill. ISBN 978-0-7006-2609-0 (cl); 978-0-7006-2610-6 (pb). Laurel Clark Shire. The Threshold of Manifest Destiny: Gender and National Expansion in Florida. Philadelphia: University of Pennsylvania Press, 2016. ix + 273 pp.; ill. ISBN 978-0-8122-4836-4 (cl). Susan Sleeper-Smith. Indigenous Prosperity and American Conquest: Indian Women of the Ohio River Valley, 1690–1792. Williamsburg, VA: Omohundro Institute of Early American History and Culture, 2018. vii + 376 pp.; ill. ISBN 978-1-4696-4058-7 (cl); 978-1-8468-591-69 (pb); 978-1-4696-4059-4 (ebook). Over the last two decades, scholars have rethought the history of empire and colonization in North America. Epitomized by Pekka Hämäläinen's Comanche Empire, historians have used the term "empire" to turn the tables on traditional assumptions about European power. They instead posit that Native empires maintained or increased control over the heart of the continent into and beyond the eighteenth century.1 At the same time, theorists of Native and Indigenous studies have used the framework of "settler colonialism" to identify the continuous drive for the erasure of multiple Native groups from North American landscapes from the eighteenth to the twenty-first centuries.2 These dual literatures open up analytical space for new considerations of the relationship between gender and empires in Native North America. If we take seriously notions of Native geopolitical power, while reckoning with the long-term eliminatory process of settler colonialism, what do we need to rethink about the histories of Indigenous women? Three new books, by Susan Sleeper-Smith, Tai S. Edwards, and Laurel Clark Shire, indicate how productive this analytical space can be. Their work shows that grounding studies deeply within women's work and culture can allow for better understandings of both Indigenous power and settler colonial processes. Tai S. Edwards's engaging study of Osage women puts both of these historiographic developments front and center. In four compact, highly readable chapters, Edwards explores women's contributions to the creation [End Page 146] of an Osage empire, and their subsequent experience of dispossession under the US empire. Ideal for undergraduate classroom use, Edwards's lucid prose takes the reader through a long sweep of Osage history. Drawing on a wealth of Indigenous sources and published anthropological research, Edwards gives a clear picture of women's lives and the function of gender complementarity at different stages of Osage history. Since she begins, rather than ends, with discussions of Two-Spirit people, who inhabited both gender roles in parallel, she prevents the reader from developing an overly-rigid sense of gender division. Instead, she shows the constant fluidity of male and female in Osage thinking. This clear summation of Osage gender norms allows Edwards to explore women's material and spiritual role in the rise of an Osage empire in the eighteenth century. Here, she relies on Kathleen DuVal's rethinking of Native power in the Arkansas River Valley and the work of such Osage specialists as Willard H. Rollings. She takes the framing of the Osage as an empire as a given, noting Hämäläinen's explication of the differences between Native American and European empires. Although Edwards agrees with Osage historians about their expansion and dominance with the coming of the fur trade, her understanding of Osage women's coequal place within this growing world offers a fresh perspective. The advent of exchanges in horses, furs, and slaves did not require a reorganization of gender roles; rather, new prosperity allowed more people to have access to spiritual knowledge, since having more goods to give away enabled more people to undergo initiation ceremonies. This meant a rise in the Osage celebration of Wa-kon'-da, the creative power, which "favored [women] with the mystic power of creating human life" (18). So, empire among the Osage, in Edwards's view, entailed a decline neither in women's spiritual importance nor in their daily work and material lives. Even Osage success in warfare and violence, a...