Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints

Jiang, Houyuan; Ralph, Daniel

doi:10.1137/s1052623497332329

Cited by 125 publications

(103 citation statements)

References 24 publications

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“…This means that after a finite number of iterations, k remains unchanged, i.e., for some k 0 and all k ≥ k 0 , k = k 0 . In this case, our smoothing method reduces to the smooth modified SQP method proposed in [9] for a smooth nonlinear optimization, see Appendix in [9]. Then Theorem A.1 in [9] indicates that d k → 0, which implies that d k ≤ k 0 for sufficiently large k. This contradicts with the above conclusion.…”

Section: Global Convergencementioning

confidence: 89%

“…Similar to reference [8], we define a modified quadratic program QP (x, u, y, , r) subproblem of (3.4) by…”

Section: A Smoothing Sqp Algorithmmentioning

confidence: 99%

“…Proof The proof process is similar to the one of Theorem A.2 in [9], just to replace the Jacobian matrix of smooth function by the generalized Jacobian V . Here we omit the detailed process of the proof.…”

Section: Global Convergencementioning

confidence: 99%

See 2 more Smart Citations

A smoothing SQP method for nonlinear programs with stability constraints arising from power systems

Tong

et al. 2010

Comput Optim Appl

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This paper investigates a new class of optimization problems arising from power systems, known as nonlinear programs with stability constraints (NPSC), which is an extension of ordinary nonlinear programs. Since the stability constraint is described generally by eigenvalues or norm of Jacobian matrices of systems, this results in the semismooth property of NPSC problems. The optimal conditions of both NPSC and its smoothing problem are studied. A smoothing SQP algorithm is proposed for solving such optimization problem. The global convergence of algorithm is established. A numerical example from optimal power flow (OPF) is done. The computational results show efficiency of the new model and algorithm.

show abstract

Section: Global Convergencementioning

confidence: 89%

“…Similar to reference [8], we define a modified quadratic program QP (x, u, y, , r) subproblem of (3.4) by…”

Section: A Smoothing Sqp Algorithmmentioning

confidence: 99%

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A smoothing SQP method for nonlinear programs with stability constraints arising from power systems

Tong

et al. 2010

Comput Optim Appl

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show abstract

“…The introduction of the smoothing parameter θ has three consequences [53]: non-smooth problems are transformed into smooth problems, except when θ=0; well-posedness can be improved in the sense that feasibility and constraint qualifications, hence stability, are often more likely to be satisfied for all values of θ; and the solvability of quadratic approximation problems is improved. Therefore, the smooth and approximate equation for the complementarity constraint is obtained as follows:…”

Section: Solution Algorithmmentioning

confidence: 99%

An MPCC Formulation and Its Smooth Solution Algorithm for Continuous Network Design Problem

Wang

2017

PROMET

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Continuous network design problem (CNDP) is searching for a transportation network configuration to minimize the sum of the total system travel time and the investment cost of link capacity expansions by considering that the travellers follow a traditional Wardrop user equilibrium (UE) to choose their routes. In this paper, the CNDP model can be formulated as mathematical programs with complementarity constraints (MPCC) by describing UE as a non-linear complementarity problem (NCP

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“…The piecewise MangasarianFromovitz constraint qualification consists of saying that constraints defining each of the branches satisfy atx the usual MFCQ. Piecewise constraint qualifications play an important role in theoretical and numerical treatment of MPEC (Jiang andRalph 2000 andFukushima 1999).…”

Section: Ifx Is a Local Solution Of Problem (13) (14) Thenmentioning

confidence: 99%

The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions

Izmailov

Solodov

2002

Mathematics of OR

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We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2-regularity (a certain kind of second-order regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2-regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to "right-hand side" perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2-regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation-based reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.1. Introduction. In this paper, we consider the problem of local approximation of a nonlinear mapping near a given point in the absence of standard combinations of smoothness and regularity assumptions. We further study consequences of this approximation, in particular those relevant for constructive description of tangent directions to level surfaces, and optimality conditions for problems with feasible regions defined by irregular mappings.Let X and Y be Banach spaces, and consider a pointx ∈ X, its neighborhood V in X, and a mapping F V → Y . In what follows, X Y stands for the space of continuous linear operators from X to Y . For a linear operator ∈ X Y , ker = x ∈ X x = 0 is its null space, im = y ∈ Y y = x for some x ∈ X is its image space, and * Y * → X * is the linear operator adjoint to , where X * and Y * denote the dual spaces of X and Y , respectively.It is well known that if F is (first-order) regular atx, i.e.,

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Smooth SQP Methods for Mathematical Programs with Nonlinear Complementarity Constraints

Cited by 125 publications

References 24 publications

A smoothing SQP method for nonlinear programs with stability constraints arising from power systems

A smoothing SQP method for nonlinear programs with stability constraints arising from power systems

An MPCC Formulation and Its Smooth Solution Algorithm for Continuous Network Design Problem

The Theory of 2-Regularity for Mappings with Lipschitzian Derivatives and its Applications to Optimality Conditions

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