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A sequential optimality condition related to the quasi-normality constraint qualification and its algorithmic consequences. (English) Zbl 1451.90166
This paper deals with optimization problems of the form $\min_{x\in X}f(x),\tag{P}$ where $$f:\mathbb{R}^{n}\longrightarrow \mathbb{R}$$ and $$X=\left\{ x\in\mathbb{R}^{n}\mid h(x) =0_{m},\,g(x) \leq 0_{p}\right\}$$, with $$h:\mathbb{R}^{n}\longrightarrow \mathbb{R}^{m}$$ and $$g:\mathbb{R}^{n}\longrightarrow \mathbb{R}^{p}$$, and $$f,g,h\in \mathcal{C}^{1}\left( \mathbb{R}^{n}\right)$$. Following the Powell-Hestenes-Rockafellar (PHR) augmented Lagrangian approach, the authors associate, with a given penalty parameter $$\rho >0$$ and Lagrange multiplier estimates $$\lambda \in \mathbb{R}^{m}$$ and $$\mu \in \mathbb{R}_{+}^{p}$$, the augmented Lagrangian function $L_{\rho}\left( x,\lambda ,\mu \right) =f(x) +\frac{\rho}{2}\left( \sum\limits_{i=1}^{m}\left( h_{i}(x) +\frac{\lambda _{i}}{\rho}\right) ^{2}+\sum\limits_{j=1}^{p}\max \left\{ 0,g_{j}(x) +\frac{\mu _{j}}{\rho}\right\} ^{2}\right).$ Each step of the PHR augmented Lagrangian algorithm for (P) consists in the minimization of $$L_{\rho}$$ for some $$\rho >0$$. Once the approximate solution is found, the penalty parameter $$\rho$$, as well as the multiplier estimates $$\lambda$$ and $$\mu$$, are updated and a new iteration starts.
An element $$x^{\ast}\in X$$ satisfies the approximate Karush-Kuhn-Tucker (AKKT) condition for (P) if there exists a sequence $$\left\{\left( x^{k},\lambda ^{k},\mu ^{k}\right) \right\} \subset \mathbb{R}^{n+m}\times \mathbb{R}_{+}^{p}$$ such that $x^{k}\longrightarrow x^{\ast},\ \nabla _{x}L\left( x^{k},\lambda^{k},\mu ^{k}\right) \longrightarrow 0_{n},\text{ and }\min \left\{-g\left(x^{k}\right) ,\mu ^{k}\right\} \longrightarrow 0_{m}, \tag{1}$ where $$L$$ denotes the ordinary Lagrangian of (P). In that case, $$x^{k}$$ is called an AKKT point and $$\left\{ x^{k}\right\}$$ an AKKT sequence. Many constraint qualifications (CQs) have been proposed to guarantee that a given AKKT point is stationary, that is, satisfies the classical Karush-Kuhn-Tucker necessary optimality conditions for $$\left( P\right)$$. Most of these CQs are based on either rank and/or positive linear dependence assumptions (among them the so-called cone continuity property, CCP in short) or pseudonormality and quasi-normality. In order to unify both types of CQs, this paper introduces a new constraint qualification called positive approximate Karush-Kuhn-Tucker (PAKKT) regularity, which consists in the existence of some element $$x^{\ast}\in X$$ and an associated sequence $$\left\{ \left( x^{k},\lambda ^{k},\mu^{k}\right) \right\} \subset \mathbb{R}^{n+m}\times \mathbb{R}_{+}^{p}$$ such that (1) holds together with $\lambda _{i}^{k}h_{i}\left( x^{k}\right) >0 \text{ if }\lim\limits_{k}\frac{\left\vert \lambda _{i}^{k}\right\vert}{\left\Vert \left( 1,\lambda^{k},\mu ^{k}\right) \right\Vert _{\infty}}>0,\tag{2}$ and $\mu _{j}^{k}g_{j}\left( x^{k}\right) >0 \text{ if }\lim\limits_{k}\frac{\left\vert \mu _{j}^{k}\right\vert}{\left\Vert \left( 1,\lambda ^{k},\mu^{k}\right) \right\Vert _{\infty}}>0.\tag{3}$ In that case, $$x^{k}$$ is said to be a PAKKT point and $$\left\{ x^{k}\right\}$$ a PAKKT sequence. The PAKKT points are AKKT points (as they satisfy (1)) whose sequences of multipliers satisfy the positivity conditions (2) and (3).
The paper shows that PAKKT-regularity is strictly weaker than both quasi-normality and CCP. It is also proved that the PHR augmented Lagrangian method converges under the new PAKKT-regularity CQ. Thus, the paper provides the first known example of a practical algorithm, the mentioned PHR augmented Lagrangian one, which converges under quasi-normality.

##### MSC:
 90C46 Optimality conditions and duality in mathematical programming 90C30 Nonlinear programming 65K05 Numerical mathematical programming methods
ALGENCAN
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