On LICQ and the uniqueness of Lagrange multipliers

Wachsmuth, Gerd

doi:10.1016/j.orl.2012.11.009

Cited by 78 publications

(66 citation statements)

References 5 publications

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“…Weaker constraint qualifications than the LICQ exist under which Theorem 1 holds [1], [28]. However, the uniqueness of Lagrange multipliers is strongly related to the LICQ and does not in general hold under weaker CQs [2], [29].…”

Section: B Optimality Conditionsmentioning

confidence: 99%

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Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

Hauswirth

Bolognani

Hug

et al. 2018

IEEE Control Syst. Lett.

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Solutions to nonlinear, nonconvex optimization problems can fail to satisfy the KKT optimality conditions even when they are optimal. This is due to the fact that unless constraint qualifications (CQ) are satisfied, Lagrange multipliers may fail to exist. Even if the KKT conditions are applicable, the multipliers may not be unique. These possibilities also affect AC optimal power flow (OPF) problems which are routinely solved in power systems planning, scheduling and operations. The complex structure -in particular the presence of the nonlinear power flow equations which naturally exhibit a structural degeneracy -make any attempt to establish CQs for the entire class of problems very challenging. In this paper, we resort to tools from differential topology to show that for AC OPF problems in various contexts the linear independence constraint qualification is satisfied almost certainly, thus effectively obviating the usual assumption on CQs. Consequently, for any local optimizer there generically exists a unique set of multipliers that satisfy the KKT conditions.

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Section: B Optimality Conditionsmentioning

confidence: 99%

“…Consequently, the genericity of LICQ is guaranteed under conditions encountered when formulating an AC OPF problem from noisy measurements, e.g., in online applications. 2 Given v 2 , θ 2 , one can always compute p 1 , p 2 , q 1 , q 2 according to (3).…”

Section: Perturbation Models For Ac Opfmentioning

confidence: 99%

Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

Hauswirth

Bolognani

Hug

et al. 2018

IEEE Control Syst. Lett.

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“…Before making the main statement we need to establish uniqueness of the optimal dual multipliers. This is done to guarantee that the multiplier drift Proof Sketch: All dual multipliers associated to the constraints in (1) can be shown to be unique using the result in [9]. Further, by writing the optimality conditions of (3) and (1) we can show that λ ⋆[k] is a linear combination of the optimal dual multipliers of (1) implying the uniqueness of…”

Section: Tracking Statementmentioning

confidence: 99%

Dynamic Power Allocation for Smart Grids Via ADMM

Maros

Jaldén

2018

2018 IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

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Electric power distribution systems encounter fluctuations in supply due to renewable sources with high variability in generation capacity. It is therefore necessary to provide algorithms that are capable of dynamically finding approximate solutions. We propose two semi-distributed algorithms based on ADMM and discuss their advantages and disadvantages. One of the algorithms computes a feasible approximate of the optimal power allocation at each time instance. We require coordination between the nodes to guarantee feasibility of each of the iterates. We bound the distance from the approximate solutions to the optimal solution as a function of the variation in optimal power allocation, and we verify our results via experiments.

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“…In particular, in the limit of an infinitely large heat bath B the total number of accessible states for B is still given by equation (18). In addition, it can be shown that the resulting value of the Lagarange multiplier, λ, is unique [17]. Hence, we can formulate a statement of the zeroth law of thermodynamics from envariance-namely, two systems S 1 and S 2 , that are in equilibrium with a large heat bath B, are also in equilibrium with each other, and they have the same temperature corresponding to the unique value of λ.…”

Section: Boltzmann's Formula For the Canonical Statementioning

confidence: 99%

Foundations of statistical mechanics from symmetries of entanglement

Deffner

Zurek

2016

New J. Phys.

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Envariance-entanglement assisted invariance-is a recently discovered symmetry of composite quantum systems. We show that thermodynamic equilibrium states are fully characterized by their envariance. In particular, the microcanonical equilibrium of a system  with Hamiltonian  H is a fully energetically degenerate quantum state envariant under every unitary transformation. The representation of the canonical equilibrium then follows from simply counting degenerate energy states. Our conceptually novel approach is free of mathematically ambiguous notions such as ensemble, randomness, etc., and, while it does not even rely on probability, it helps to understand its role in the quantum world.

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On LICQ and the uniqueness of Lagrange multipliers

Cited by 78 publications

References 5 publications

Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

Dynamic Power Allocation for Smart Grids Via ADMM

Foundations of statistical mechanics from symmetries of entanglement

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