Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear $\cal K$-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone $\cal K$. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in ${\cal O}(\log(1/\rho)/\rho^{3})$ inner iterations, where $\rho>0$ is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method

Iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian method based on the classical Lagrangian function and a full Lagrange multiplier update

This paper establishes the iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly-constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in ${\cal O}(\rho^{-3})$ ACG iterations (up to a logarithmic term) where $\rho>0$ is the tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL

The OSU1/QUA2/TSD2-Encoded Putative Methyltransferase Is a Critical Modulator of Carbon and Nitrogen Nutrient Balance Response in Arabidopsis

The balance between carbon (C) and nitrogen (N) nutrients must be tightly coordinated so that cells can optimize their opportunity for metabolism, growth and development. However, the C and N nutrient balance perception and signaling mechanism remains poorly understood. Here, we report the isolation and characterization of two allelic oversensitive to sugar1 mutants (osu1-1, osu1-2) in Arabidopsis thaliana. Using the cotyledon anthocyanin accumulation and root growth inhibition assays, we show that the osu1 mutants are more sensitive than wild-type to both of the imbalanced C/N conditions, high C/low N and low C/high N. However, under the balanced C/N conditions (low C/low N or high C/high N), the osu1 mutants have similar anthocyanin levels and root lengths as wild-type. Consistently, the genes encoding two MYB transcription factors (MYB75 and MYB90) and an Asn synthetase isoform (ASN1) are strongly up-regulated by the OSU1 mutation in response to high C/low N and low C/high N, respectively. Furthermore, the enhanced sensitivity of osu1-1 to high C/low N with respect to anthocyanin accumulation but not root growth inhibition can be suppressed by co-suppression of MYB75, indicating that MYB75 acts downstream of OSU1 in the high C/low N imbalance response. Map-based cloning reveals that OSU1 encodes a member of a large family of putative methyltransferases and is allelic to the recently reported QUA2/TSD2 locus identified in genetic screens for cell-adhesion-defective mutants. Accumulation of OSU1/QUA2/TSD2 transcript was not regulated by C and N balance, but the OSU1 promoter was slightly more active in the vascular system. Taken together, our results show that the OSU1/QUA2/TSD2-encoded putative methyltransferase is required for normal C/N nutrient balance response in plants