A globally convergent, implementable multiplier method with automatic penalty limitation

Polak, E.; Tits, A.L.

doi:10.1007/bf01442901

Cited by 25 publications

(4 citation statements)

References 19 publications

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“…Penalization has been introduced by Courant [Cou43] and it is very common in optimization theory (see e.g. [Cea78] and [PT80]). Then we prove some a priori estimates for the penalized (ε-regularized) solution, and finally pass to the limit as ε → 0 to end up with the existence of the solution for the initial (non regularized) problem.…”

Section: The Source Termsmentioning

confidence: 99%

The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

et al. 2018

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In this article we propose a new formulation of the equations of the humid atmosphere with a multi-phase saturation generalizing thus the model introduced in [TWu15] and [TWa15]. More precisely, we consider the more realistic situation where the humid atmosphere comprises these components, namely water vapor, liquid water and cloud condensates and furthermore the saturation concentration is not constant. With the additional constraint that the vapor mass ratio q v is less than the saturation concentration q vs , which depends itself on the state, we are led, from the mathematical point of view, to introduce and handle a system of equations and inequations involving some quasi-variational inequalities for which we prove the existence of solutions.

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Section: The Source Termsmentioning

confidence: 99%

The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

et al. 2018

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“…When the constraint q v ≤ q vs itself depends on the solution T , the authors are led to introduce and handle a system of equations and inequations involving a so-called quasi-variational inequality for which they prove the existence of solutions using penalization techniques. Penalization has been introduced by R. Courant [19] and it is very common in Optimization Theory (see e.g., [15] and [53]). For general results on (quasi-)variational inequalities and their utilization in economics, mechanics and physics, see among a vast literature [12,2,6,7,5,8,9,4,22,23,26,41,36,38,44,47,50,49,48].…”

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confidence: 99%

Mathematical analysis of a cloud resolving model including the ice microphysics

Cao¹,

Jia²,

Témam³

et al. 2021

Discrete &Amp; Continuous Dynamical Systems - A

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We extend our study for the warm cloud model in [13] to the analysis of a more general cloud model including the ice microphysics in [28]. The moisture variables comprise water vapor, cloud condensates (cloud water, cloud ice), and cloud precipitations (rain, snow), with respective mass ratios qv, qc and qp. A typical assumption in [13] for the calculation of condensation rate is that the warm clouds are exactly at water saturation with no supersaturation in general. When the ice microphysics are included, the situation becomes more complicated. We have to consider both the saturation mixing ratio with respect to water (qvw) and the saturation with respect to ice (q vi) when the temperature T is below the freezing point Tw but above the threshold T i for homogeneous ice nucleation. A remedy, acceptable from the physical and mathematical viewpoints, is to define the overall saturation mixing ratio qvs as a convex combination of qvw and q vi. Under this setting, supersaturation can still be avoided and we have the constraint qv ≤ qvs with qvs depending itself on the state. Mathematically, we are led to a system of equations and inequations involving some quasi-variational inequalities for which we prove the global existence and regularity of solutions.

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“…While the adaptive selection of the penalty parameter is often heuristic, in some contexts, authors have proposed formal adaptation rules that guarantee that an appropriate value of the parameter will eventually be obtained and will be kept for the remainder of the solution process; this goes back several decades (e.g., [30] as well as, in the context of augmented Lagrangian, [17,35]) and also includes more recent work such as [11,39]. Finally, in the past two decades, exact penalty functions have been used successfully in the solution of mathematical programs with complementary constraints (MPCC), e.g., [16,32] and references therein.…”

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confidence: 99%

An infeasible-start framework for convex quadratic optimization, with application to constraint-reduced interior-point and other methods

Laiu

Tits

2021

Math. Program.

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A framework is proposed for solving general convex quadratic programs (CQPs) from an infeasible starting point by invoking an existing feasible-start algorithm tailored for inequality-constrained CQPs. The central tool is an exact penalty function scheme equipped with a penalty-parameter updating rule. The feasible-start algorithm merely has to satisfy certain general requirements, and so is the updating rule. Under mild assumptions, the framework is proved to converge on CQPs with both inequality and equality constraints and, at a negligible additional cost per iteration, produces an infeasibility certificate, together with a feasible point for an (approximately) 1 -least relaxed feasible problem, when the given problem does not have a feasible solution. The framework is applied to a feasible-start constraint-reduced interior-point algorithm previously proved to be highly performant on problems with many more inequality

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A globally convergent, implementable multiplier method with automatic penalty limitation

Cited by 25 publications

References 19 publications

The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality

Mathematical analysis of a cloud resolving model including the ice microphysics

An infeasible-start framework for convex quadratic optimization, with application to constraint-reduced interior-point and other methods

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