Active Set Strategy Research Articles

New approach for the nonlinear programming with transient stability constraints arising from power systems

This paper presents a new approach for solving a class of complicated nonlinear programming problems arises from optimal power flow with transient stability constraints (denoted by OTS) in power systems. By using a functional transformation technology proposed in Chen et al. (IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48:327---339, [2001]), the OTS problem is transformed to a semi-infinite programming (SIP). Then based on the KKT (Karush-Kuhn-Tucker) system of the reformulated SIP problem and the finite approximation technology, an iterative method is presented, which develops Wu-Li-Qi-Zhou' (Optim. Methods Softw. 20:629---643, [2005]) method. In order to save the computing cost, some typical computing technologies, such as active set strategy, quasi-Newton method for the subproblems coming from the finite approximation model, are addressed. The global convergence of the proposed algorithm is established. Numerical examples from power systems are tested. The computing results show the efficiency of the new approach.

Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing

Our goal is to identify the volatility function in Dupire’s equation from given option prices. Following an optimal control approach in a Lagrangian framework, a globalized sequential quadratic programming (SQP) algorithm combined with a primal-dual active set strategy is proposed. Existence of local optimal solutions and of Lagrange multipliers is shown. Furthermore, a sufficient second-order optimality condition is proved. Finally, some numerical results are presented underlining the good properties of the numerical scheme.

An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems

We present a new active-set strategy which can be used in conjunction with exponential (entropic) smoothing for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. The main effect of the active-set strategy is to dramatically reduce the number of gradient calculations needed in the optimization. Discretization of multidimensional domains gives rise to minimax problems with thousands of component functions. We present an application to minimizing the sum of squares of the Lagrange polynomials to find good points for polynomial interpolation on the unit sphere in ℝ3. Our numerical results show that the active-set strategy results in a modified Armijo gradient or Gauss-Newton like methods requiring less than a quarter of the gradients, as compared to the use of these methods without our active-set strategy. Finally, we show how this strategy can be incorporated in an algorithm for solving semi-infinite minimax problems.

On the Convergence of the Pshenichnyi-Pironneau-Polak Minimax Algorithm with an Active Set Strategy

The purpose of this technical note is to present a proof of convergence of the Pshenichnyi-Pironneau-Polak (PPP) minimax algorithm (see Algorithm 2.4.1 in Polak, Optimization: Algorithms and Consistent Approximations, Springer, [1997]), modified to use an active set strategy. This active set strategy was formally derived in Polak (Optimization: Algorithms and Consistent Approximations, Springer, [1997]) from those used in the methods of feasible directions developed by Zoutendijk (Methods of Feasible Directions, Elsevier, [1960]) and Polak (Computational Methods in Optimization: A Unified Approach, Academic, [1971]). The resulting e-Active PPP algorithm was presented as Algorithm 2.4.34, in Polak (Optimization: Algorithms and Consistent Approximations, Springer, [1997]), without any proofs.

A mortar based contact formulation for non-linear dynamics using dual Lagrange multipliers

Many existing algorithms for the analysis of large deformation contact problems use a so-called node-to-segment approach to discretize the contact interface between dissimilar meshes. It is well known, that this discretization strategy may lead to problems like loss of convergence or jumps in the contact forces. Additionally it is popular to use penalty methods to satisfy the contact constraints. This necessitates a user defined penalty parameter the choice of which is somehow arbitrary, problem dependent and might influence the accuracy of the analysis. In this work, a frictionless segment-to-segment contact formulation is presented that does not require any user defined parameter to handle the non-linearity of the contact conditions. The approach is based on the mortar method enforcing the compatibility condition along the contact interface in a weak integral sense. The application of dual spaces for the interpolation of the Lagrange multiplier allows for a nodal decoupling of the contact constraints. A local basis transformation in combination with a primal–dual active set strategy enables the exact enforcement of the contact constraints via prescribed incremental boundary conditions. Due to the biorthogonality condition of the basis functions the Lagrange multipliers can be locally eliminated. A static condensation leads to a reduced system of equations to be solved solely for the unknown nodal displacements. Thus the size of the system of equations remains constant during the whole calculation. The discrete Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For the analysis of dynamic contact problems the proposed contact description is combined with the implicit Generalized Energy-Momentum Method. Several numerical examples illustrate the performance of the suggested contact formulation.

Uniqueness for solutions of differential complementarity problems

In this paper we consider the question of which matrices M give unique solutions for Differential Complementarity Problems (Mandelbaum 1989, unpublished manuscript) of the form $$\begin{array}{ll}&\frac{dw}{dt} = M\, z + q(t),\quad w(0) = w_{0},\\ K \ni&z(t) \perp w(t) \in K^{*} \quad {\rm for\,all}\,t,\end{array}$$ with applications to complementarity and discretized l1-regularization problems. Assuming semi-smoothness it is shown that super-linearly convergent Newton methods can be globalized, if appropriate descent directions are used for the merit function |F(x)|2. Special attention is paid to directions obtained from the primal-dual active set strategy.

Energy-Consistent CoRotational Schemes for Frictional Contact Problems

In this paper, we consider the unilateral frictional contact problem of a hyperelastic body in the case of large displacements and small strains. In order to retain the linear elasticity framework, we decompose the deformation into a large global rotation and a small elastic displacement. This corotational approach is combined with a primal-dual active set strategy to tackle the contact problem. The resulting algorithm preserves both energy and angular momentum.

A semi‐smooth Newton method for an inverse problem in option pricing

AbstractWe present an optimal control approach using a Lagrangian framework to identify local volatility functions from given option prices. We employ a globalized sequential quadratic programming (SQP) algorithm and implement a line search strategy. The linear‐quadratic optimal control problems in each iteration are solved by a primal‐dual active set strategy which leads to a semi‐smooth Newton method. We present first‐ and second‐order analysis as well as numerical results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scalable algorithm for solving 3D contact problems with friction

AbstractA scalable algorithm for solving contact problems with Coulomb friction is presented. It combines fixed point iterations with an augmented Lagrangian loop and an active set strategy. Numerical experiments indicate scalability. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On two numerical methods for state-constrained elliptic control problems

A linear-quadratic elliptic control problem with pointwise box constraints on the state is considered. The state constraints are treated by a Lavrentiev type regularization. It is shown that the Lagrange multipliers associated with the regularized state constraints are functions in L 2. Moreover, the convergence of the regularized controls is proven for regularization parameter tending to zero. To solve the problem numerically, an interior point method and a primal–dual active set strategy are implemented and treated in function space.

On a semi-smooth Newton method and its globalization

This paper addresses the globalization of the semi-smooth Newton method for non-smooth equations F(x) = 0 in $${\mathbb{R}}^m$$ with applications to complementarity and discretized l1-regularization problems. Assuming semi-smoothness it is shown that super-linearly convergent Newton methods can be globalized, if appropriate descent directions are used for the merit function |F(x)|2. Special attention is paid to directions obtained from the primal-dual active set strategy.

ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems: a corrigendum

ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems: a corrigendum

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].

An active set strategy to generate quadrilateral grids

We consider the problem of the generation of quadrilateral grids on planar domains. This problem is numerically solved by a two phases method: an iterative procedure based on the well-known variational approach, and an active set procedure to obtain unfolded quadrilaterals. This second phase is performed only when it is really necessary, in fact the first phase alone gives satisfactory results on a large number of domains. This two phases approach provides a robust method with low computational cost. Numerical experiments show that this method is able to generate unfolded grids also on complex domains.

An online active set strategy to overcome the limitations of explicit MPC

AbstractNearly all algorithms for linear model predictive control (MPC) either rely on the solution of convex quadratic programs (QPs) in real time, or on an explicit precalculation of this solution for all possible problem instances. In this paper, we present an online active set strategy for the fast solution of parametric QPs arising in MPC. This strategy exploits solution information of the previous QP under the assumption that the active set does not change much from one QP to the next. Furthermore, we present a modification where the CPU time is limited in order to make it suitable for strict real‐time applications. Its performance is demonstrated with a challenging test example comprising 240 variables and 1191 inequalities, which depends on 57 parameters and is prohibitive for explicit MPC approaches. In this example, our strategy allows CPU times of well below 100 ms per QP and was about one order of magnitude faster than a standard active set QP solver. Copyright © 2007 John Wiley & Sons, Ltd.

ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transformed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The method is compared to alternative approaches proposed in the literature.

A stable energy‐conserving approach for frictional contact problems based on quadrature formulas

AbstractA common approach for the numerical simulation of non‐linear multi‐body contact problems is the use of Lagrange multipliers to model the contact conditions. The stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes. By this, the spurious algorithmic oscillations in the multiplier do not occur any more, which facilitates the application of the primal–dual active set strategy to dynamical contact problems. The new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost. In addition the conservation properties of the underlying algorithm are transferred to the modified mass version. Different numerical examples for frictional two‐body contact problems illustrate the improvement in the results for the contact stresses. Copyright © 2007 John Wiley & Sons, Ltd.

A trust-region method by active-set strategy for general nonlinear optimization

The paper explores a trust-region active-set algorithm for general nonlinear optimization with nonlinear equality and inequality constraints. In this algorithm, an active-set strategy is used together with trust-region methods to compute the trial step. L 1 penalty functions are employed to obtain the global convergence. The global convergence of this algorithm is proved under standard conditions. The numerical tests show the efficiency of the proposed algorithm.

An efficient algorithm for state estimation problems with coupling inequality constraints

In this paper, we propose an efficient duality based algorithm to solve the state estimation problems with coupling inequality constraints. The proposed algorithm is imbedded with two techniques, the first is the parallel block scaled gradient technique which can solve the unconstrained minimization problem induced by the unconstrained dual problem in the solution iterative process, the second one is the modified Active set strategy based DPPQN technique can solve the disjoint small projection problems induced by the coupling linear constraints efficiently. We combine these two techniques with a Projected Jacobin method possessing decomposition effects. With the decomposition, our algorithm is computationally efficient. We test our method in solving numerous examples of state estimation problems with coupling inequality constraints in IEEE 118-bus system. The simulation results show that the proposed algorithm is efficient in solving the problems and successfully handle the coupling inequality constraints. Furthermore, the computational efficiency of the proposed algorithm is more significant while the numbers of coupling inequality constraints and/or inequality constraints are increased.

AN ONLINE ACTIVE SET STRATEGY FOR FAST ADJOINT BASED NONLINEAR MODEL PREDICTIVE CONTROL

Although nonlinear model predictive control (NMPC) has become a well-established control approach its application to fast systems with millisecond timescales is still a challenge. In order to reduce the online computational effort significantly, we propose to combine two recently developed techniques: the adjoint based real-time iteration scheme which allows treatment of nonlinear processes while keeping constraint sensitivities fixed and the online active set strategy for fast solution of the resulting parametric quadratic program. Considerable performance enhancement compared with the standard real-time iteration scheme is demonstrated by controlling a system with fast dynamics.