The Expanded Lagrangian System for Constrained Optimization Problems

Poore, Aubrey B.; Al-Hassan, Qassem M.

doi:10.1137/0326024

Cited by 23 publications

(5 citation statements)

References 10 publications

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“…More recently, probability-one homotopy methods have been applied to solving optimization problems, such as optimal control [39], [40] and statistical learning [41]. Typically, the homotopy methods in optimization focus on parametrizing the firstorder optimality conditions [42], [43] or the objective function ( [44], [45]). Homotopy methods have also been applied in the field of power systems, primarily to solve the power flow (PF) problem for cases that do not converge [46], [47], [48], [49], [50].…”

Section: Homotopy For Optimizationmentioning

confidence: 99%

An Efficient Homotopy Method for Solving the Post-Contingency Optimal Power Flow to Global Optimality

et al. 2022

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Optimal power flow (OPF) is a fundamental problem in power systems analysis for determining the steady-state operating point of a power network that minimizes the generation cost. In anticipation of component failures, such as transmission line or generator outages, it is also important to find optimal corrective actions for the power flow distribution over the network. The problem of finding these post-contingency solutions to the OPF problem is challenging due to the nonconvexity of the power flow equations and the large number of contingency cases in practice. In this paper, we introduce a homotopy method to solve for the post-contingency actions, which involves a series of intermediate optimization problems that gradually transform the original OPF problem into each contingency-OPF problem. We show that given a global solution to the original OPF problem, a global solution to the contingency problem can be obtained using this homotopy method, under some conditions. With simulations on Polish and other European networks, we demonstrate that the effectiveness of the proposed homotopy method is dependent on the choice of the homotopy path and that homotopy yields an improved solution in many cases. For at least 5% of the test cases, bad local minima were identified, and the homotopy method yielded a solution that was significantly better than state-of-the-art interior point methods in terms of reducing the violation cost during a catastrophic contingency scenario.INDEX TERMS Power systems, optimal power flow, nonconvex optimization, contingency analysis I. INTRODUCTIONRecently, there has been elevated interest in studying the

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Section: Homotopy For Optimizationmentioning

confidence: 99%

An Efficient Homotopy Method for Solving the Post-Contingency Optimal Power Flow to Global Optimality

et al. 2022

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“…The expanded Lagrangian homotopy method of Poore [30], [31] is applicable to the general nonlinear programming problem min O(x) subject to g(x) ~< I).…”

Section: Expanded Lagrangian Homotopymentioning

confidence: 99%

Globally convergent homotopy algorithms for nonlinear systems of equations

Watson

1990

Nonlinear Dyn

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Abstract. Probability-one homotopy methods are a class ofalgorithnls for solving nonlinear svstems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely. These new globally convergent homotopy techniques have been successfully applied to solve Brouwer fixed point problems, polynomial systems of equations, constrained and unconstrained optimization problems, discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements, and finite difference, collocation, and Galerkin approximations to nonlinear partial differential equations. This paper introduces, in a tutorial fashion, the theory of globally convergent homotopy algorithms, describes some computer algorithms and mathematical software, and presents several nontrivial engineering applications.

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“…Since then the development of homotopy methods (and interior point methods, which can be viewed as a variant of homotopy methods) in optimization has blossomed- [20,42,47,48,53,54,55,58], just to mention a few homotopy references. The application of classical continuation, homotopy algorithms, and probability-one homotopy algorithms (see [20] or [55] for a discussion of the distinction between these three) to linear complementarity problems was thoroughly explored in [57], based on the theory in [52,56].…”

Section: Introductionmentioning

confidence: 99%

A Globally Convergent Probability-One Homotopy for Linear Programs with Linear Complementarity Constraints

Watson¹,

Billups²,

Mitchell³

et al. 2013

SIAM J. Optim.

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A solution of the standard formulation of a linear program with linear complementarity constraints (LPCC) does not satisfy a constraint qualification. A family of relaxations of an LPCC, associated with a probability-one homotopy map, proposed here is shown to have several desirable properties. The homotopy map is nonlinear, replacing all the constraints with nonlinear relaxations of NCP functions. Under mild existence and rank assumptions, (1) the LPCC relaxations RLPCC(λ) have a solution for 0 ≤ λ ≤ 1; (2) RLPCC(1) is equivalent to LPCC; (3) the Kuhn-Tucker constraint qualification is satisfied at every local or global solution of RLPCC(λ) for almost all 0 ≤ λ < 1; (4) a point is a local solution of RLPCC(1) (and LPCC) if and only if it is a Kuhn-Tucker point for RLPCC(1); and (5) a homotopy algorithm can find a Kuhn-Tucker point for RLPCC(1). Since the homotopy map is a globally convergent probability-one homotopy, robust and efficient numerical algorithms exist to find solutions of RLPCC(1). Numerical results are included for some small problems.

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The Expanded Lagrangian System for Constrained Optimization Problems

Cited by 23 publications

References 10 publications

An Efficient Homotopy Method for Solving the Post-Contingency Optimal Power Flow to Global Optimality

An Efficient Homotopy Method for Solving the Post-Contingency Optimal Power Flow to Global Optimality

Globally convergent homotopy algorithms for nonlinear systems of equations

A Globally Convergent Probability-One Homotopy for Linear Programs with Linear Complementarity Constraints

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