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In this paper, we consider the generalized nonlinear complementarity problem in engineering modeling (GNCP). We propose a solution method to solve the GNCP, and its global convergence is also established under weaker conditions.

A new scheme for incorporating generator Q limit adjustments automatically in Newton Raphson (NR) load flow method is proposed. The current practice for handling this feature is to adopt bus type switching method after the solution (without these limits) converges to a reasonable extent. However, the precise details of this scheme are not well documented and they vary across implementations. A modified Newton-Raphson load flow method is proposed here which includes this adjustment feature at the problem formulation stage itself. In the proposed approach, the limit constraints at the generator buses are viewed as a set of complementary relations. A scheme is evolved to combine these complementary relations at the generator buses into a single equation and use it along with the real power equation of the corresponding generator bus by using the Nonlinear Complementarity Problem (NCP) framework. This scheme though similar to the Mixed Complementarity Problem (MCP) approach proposed recently for this problem, is basically different. This modified formulation in the complementarity framework succeeds in eliminating many of the ambiguities inherent in the earlier MCP formulation and exhibits high level of consistency in its performance. The proposed method has been extensively tested and its effectiveness has been compared with bus type switching scheme as well as the MCP approach through simulation studies.

A new augmented Lagrangian Multiplier method with nonlinear complementarity problem (NCP) function and a Lagrangian multiplier method are proposed for the minimization of equation and inequality constraints. This method is based on the solutions of the unconstrained optimization which is a reformulation of the primal constrained problem. The global convergence can also be obtained under mild conditions.

Based on the smoothing NCP function, we first reformulate the generalized nonlinear complementarity problem over a polyhedral cone as a smoothing system of equations, and then propose a new smoothing inexact Newton method for solving it. In each iteration, the corresponding linear system is solved only inexact solution. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the generalized nonlinear complementarity problem. For the proposed method, we also obtain its global convergence under weaker conditions, and we further establish its local super linear(quadratic) convergence under the BD-regular assumption.

Constrained optimization problems play a significant role within optimization problems. In this paper, a novel method, decomposed unlimited point method (DUPM), is proposed to modify the Karush-Kuhn-Tucker (KKT) condition of constrained optimization problems. In the DUPM, KKT condition can be transformed into equations without any limitation in the variable space. Afterwards, the equivalent equations are solved by Levenberg-Marquardt method (LMM), which is the first attempt ever of applying LMM to such situations. Simulation results on various numerical examples demonstrate that DUPM is able to transform the primal KKT condition into equations without changing the functions' characteristics such as continuity and smoothness unlike nonlinear complementarity problem method (NCPM), and LMM can be widely used to solve the equivalent equations with a quadratic convergence rate.

The problem of computing economic equilibria can be considered as a nonlinear complementarity problem. A globally and superlinearly convergent smoothing method for solving nonlinear complementarity problem is presented, analyzed and applied on several economic equilibria test problems with known equilibria.

Empirical research shows that the call provision embedded in the convertible bonds and American call options may be delayed for different reasons. While there are several researches focused on recall provision embedded in the callable American call options under intensity-based framework, few performed detailed research on that in convertible bonds. Therefore, the main purpose of the current work is to model and study the call provision embedded in the callable convertible bonds using intensity-based approach. We formulate this as an optimal stopping problem. Under certain assumptions, we can then get a nonlinear complementarity problem that can be solved by numerical procedures. We also consider credit risk. Since there are already lots of literatures on this, we only provide a theoretical formulation under the AFV model framework.

International audience; To plan a robotic task involving intermittent contact, such as an assembly task, it is helpful to be able to simulate the task accurately and efficiently. In the past ten years, the prox formulation of the equations of motion has arisen as a competitive alternative to the well-known linear and nonlinear complementarity problem (LCP and NCP) formulations. In this paper, we compare these two formulations, showing through a set-based argument that the formulations are equivalent. Second, we provide simple examples to compare the most common approaches for solving these formulations. The prox formulation is solved by fixed-point iteration while the complementarity formulation is solved by a pivoting scheme, known as Lemke's algorithm. The well-known paradox of PAINLEVÉ is used in a case where two solutions exist to illustrate that the fixed-point scheme can fail while the pivoting scheme will succeed.

A smoothing function is used to reformulate the P * nonlinear complementarity problem (P * NCP) as a parameterized smooth equation which can be solved by a non-interior-point smoothing method. Under an assumption that the P * NCP has a nonempty solution set, the proposed algorithm is proved to be globally convergent and the iteration sequence generated by the proposed algorithm is bounded. Furthermore, this assumption is weaker than the ones used in most existing smoothing algorithms. Without any additional assumption, the solution obtained by the proposed algorithm is shown to be a maximally complementary solution of the P * NCP.

A new primal-dual interior-point algorithm for P ∗ (κ) nonlinear complementarity problems is proposed. New search directions and proximity functions are proposed based on a simple kernel function which is neither a logarithmic barrier nor a self-regular function. We show that if a strictly feasible starting point is available, then the algorithm have the favorable polynomial complexity.

In last decades, there has been much effort on the solution and the analysis of the nonlinear complementarity problem (NCP) by reformulating NCP as an unconstrained minimization. In this paper, we propose a new method for the NCP under the condition that the solution exists. We formulate the NCP into a multiobjective optimization problem (MOP), and define zero-efficient solution of MOP. Then we indicate that zero-efficient solution of the MOP is also the solution to the NCP. Finally we use minimax method for the MOP. Numerical results indicate that the proposed method is promising.

This paper presents a new method for solving three-dimensional frictional contact problems. A semismooth Newton method for solving nonlinear complementarity problems is proposed based on generalized Fischer-Burmeister functions. The parametric variational principle and parametric quadratic programming method are applied to the analysis of the three-dimensional frictional contact problem. The solution of the contact problem is finally reduced to a linear complementarity problem. Then the semismooth Newton method presented is employed to solve the problem. Results of numerical experiments demonstrate the efficiency and reliability of the method proposed.

In this paper we study predictor-corrector smoothing Newton method which were designed by Z. H. Huang, J. Han, and Z. Chen for nonlinear complementarity problem, we extends the Algorithm to second-order cone complementarity (SOCCP). Based on the Chen and Mangasarian smoothing function, we present a predictor-corrector smoothing Newton algorithm for solving the SOCCP. The neighbourhood of the path does not appear in the Algorithm. Thus, it does not need a few additional computations which keep the iteration sequence staying in the given neighbourhood. The algorithm is simpler than a predictor-corrector smoothing method by Chi Xiaoni, Liu Sanyang and this algorithm does not have restrictions regarding its starting point. The globally and locally superlinearly convergent under suitable assumptions are shown. Some preliminary computational results are reported.

In this paper, the electric power supply chain has multiple players, such as fuel suppliers, power generators and customers, each maximizing their individual profit. Coupling the optimal decision-making of chain members, a model of government control with CO2 emission tax in electric power supply chain is built on the basis of the Cournot game. This model shows the government's participation in and out of the game. The model is a multi-layer optimization model, whose KKT system is a mixed nonlinear complementarity problem. By using the nonlinear complementarity function to reformulate the KKT system into a set of algebraic equations, this model can be solved simply and fast. Finally, corresponding to the different operational conditions of power network, the Nash equilibrium and numerical results of this model is analyzed, and the optimal CO2 emission tax is obtained to achieve the maximization social benefits. Case studies show the successful application of the proposed model.

In this paper, we consider nonlinear complementarity problem in contact mechanics modeling (NCP). To solve the problem, we first establish a global error bound for NCP, and then propose a new type of solution method to solve the NCP based on the error bound estimation. The global convergence is also established, and the given numerical experiments shows the efficiency of the method.

From the perspective of entire supply chain of electric power with fuel supply, power generation, transmission and consumption, how to operate and coordinate the power enterprises is an important research topic. Coupling the optimal decision-making of chain members, including fuel suppliers, power generators, power suppliers or customers and independent system operator, the Nash equilibrium model of electric power supply chain is proposed with satisfying the constraints of power network. This model is a multi-layer optimization model, whose KKT system is a mixed nonlinear complementarity problem. By using the nonlinear complementarity function to reformulate the KKT system into a set of algebraic equations, the Nash equilibrium of electric power supply chain is solved simply and fast. Finally, corresponding to different operational conditions of power network, the Nash Equilibrium of electric power supply chain is analyzed. It is indicated that the fuel suppliers and power generators are favorable to joint decision-making for obtaining more profits.

This paper considers user equilibrium as a nonlinear complementarity problem when the Jacobian of arc travel cost function is symmetric and asymmetric. It shows solutions of user equilibrium exist and corresponding arc flow is unique. It also presents a novel modified FBLSA algorithm which makes full use of the advantage of dealing with large-size road network of column generation method.Finally,numerical results show this algorithm successfully decreases the number of variables and enjoys quick rate of convergence.

In this paper we propose a new potential reduction interior-point method for monotone nonlinear complementarity problem, which based on the wide-neighborhood Ninfin -(beta). Our method is ageneralization of Mizuno, Todd and Ye's result. Although the search direction of it is the same as that of the path-following algorithm, the step size is determined as the minimum point of the potential function in the neighborhood. Therefore, the duality gap is reduced by a fixed positive constant at each step. Finally, the polynomial complexity O(nt) is attained when the problem satisfies a scaled Lipschitz condition, where t is a positiveconstant.

A semismooth subdifferential method (SSM) for solving optimal power flow (OPF) is presented. The Karush-Kuhn-Tucker (KKT) conditions of OPF system, by introducing nonlinear complementarity problem (NCP) function, are transformed equivalently to a set of semismooth nonlinear algebraic equations. Then an inexact Levenberg-Marquardt (L-M) algorithm based on the subdifferential is improved and employed to solve the equations. The positive definitiveness of the iterative coefficient matrix is enhanced by using L-M parameter, while a reformed nonmonotone line search is used to enforce global convergence. Finally, the effectiveness of the SSM is shown by analyzing the computational performance and convergence characteristic of the proposed algorithm on test systems.

To accurately simulate the dynamic decisionmaking of market participants, this paper presents a dynamic bidding model of power markets considering power network transmission constraints based on the supply function and Nash equilibrium theory. The new model is composed of a differential dynamic system and semi-smooth equations system which are reformulated by nonlinear complementarity problem (NCP). The bidding variable is considered as the linear and quadratic coefficients of the supply function. Based on this model, as an example with six lines and five buses, the Nash equilibrium was analyzed in different market parameters and different operational conditions of transmission network: non-congestion and congestion; by using the numerical simulation approach, the effect of different market parameters to the dynamic behaviors was discussed; and the results were compared with the Cournot model. The simulating results show that the new model is valid.

This paper studies an equivalent traffic network equilibrium model with elastic demands for supply chain network equilibrium model. The equivalent model is formulated as a nonlinear complementarity problem. A Levenberg-Marquardt-type method was used to solve this problem. Numerical example demonstrates that the method can solve the equivalent traffic network equilibrium model more efficient than to solve the supply chain network equilibrium model.

In the deregulated environment, optimization tools that coordinate both system and security and economy are widely used to support system operation and decision. In this paper, we present a new decoupled approach to solve optimal power flow (OPF) and available transfer capability (ATC) problems. First, the KKT system of original optimization problem is reformulated equivalently to nonsmooth equations using a so-called nonlinear complementarity problem (NCP) function. Based on the new reformulation, we then apply the inherent weak-coupling characteristics of power systems and design a decoupled Newton algorithm in which a decomposition-correction strategy is considered. The combination of semismooth Newton method and decoupled strategy shares those advantages of two methods such as fast convergence and saving computation cost. Meanwhile, the new method is supported by theoretical convergence. Numerical examples of both OPF and ATC problems demonstrate that the new algorithm has potential to solve large-scale optimization problems.

This paper presents the semismooth Newton-type algorithms for solving optimal power flow (OPF) problems. Considering that there exist plentiful bounded constraints in OPF, the paper treats general inequality constraints and bounded constraints separately. By introducing a diagonal matrix and the nonlinear complementarity function, the Karush-Kuhn-Tucker (KKT) system of the OPF is transformed equivalently to a system of nonsmooth bounded constrained equations. Comparing with the classical OPF methods and the nonlinear complementarity problem (NCP) method, this treatment has two remarkable advantages. First, it has a strong ability to handle the inequality constraints in OPF problems. Second, it reduces the number of dual variables in the KKT system. Based on the reformulated equations, the paper designs a projected semismooth Newton algorithm which has nice global and local convergence property. Furthermore, according to the weak-coupling property in power systems, the paper presents a decoupled semismooth Newton-type algorithm. The decoupled method solves the system of equations via solving two lower dimension problems. Therefore the method saves the computing cost in theory. Some standard IEEE systems are used to test the two algorithms, and numerical results show that the proposed algorithms are promising for solving OPF problems

We examine how the selfish behavior of heterogeneous users in a network can be regulated through economic disincentives, i.e., through the introduction of appropriate taxation. One wants to impose taxes on the edges so that any traffic equilibrium reached by the selfish users who are conscious of both the travel latencies and the taxes will minimize the social cost, i.e., will minimize the total latency. We generalize previous results of Cole, Dodis and Roughgarden that held for a single origin-destination pair to the multicommodity setting. Our approach, which could be of independent interest, is based on the formulation of traffic equilibria as a nonlinear complementarity problem by Aashtiani and Magnanti (1981), We extend this formulation so that each of its solutions will give us a set of taxes that forces the network users to conform, at equilibrium, to a certain prescribed routing. We use the special nature of the prescribed minimum-latency flow in order to reduce the difficult nonlinear complementarity formulation to a pair of primal-dual linear programs. LP duality is then enough to derive our results.