Modified Augmented Lagrange Multiplier Methods for Large-Scale Chemical Process Optimization

Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can be applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale problems.

Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.

Active set truncated-Newton algorithm for simultaneous optimization of distillation column

An active set truncated-Newton algorithm (ASTNA) is proposed to solve the large-scale bound constrained sub-problems. The global convergence of the algorithm is obtained and two groups of numerical experiments are made for the various large-scale problems of varying size. The comparison results between ASTNA and the subspace limited memory quasi-Newton algorithm and between the modified augmented Lagrange multiplier methods combined with ASTNA and the modified barrier function method show the stability and effectiveness of ASTNA for simultaneous optimization of distillation column.

A Local Deviation Constraint Based Non-Rigid Structure From Motion Approach

In many traditional non-rigid structure from motion(NRSFM)approaches,the estimation results of part feature points may significantly deviate from their true values because only the overall estimation error is considered in their models.Aimed at solving this issue,a local deviation-constrained-based column-space-fitting approach is proposed in this paper to alleviate estimation deviation.In our work,an effective model is first constructed with two terms:the overall estimation error,which is computed by a linear subspace representation,and a constraint term,which is based on the variance of the reconstruction error for each frame.Furthermore,an augmented Lagrange multipliers(ALM)iterative algorithm is presented to optimize the proposed model.Moreover,a convergence analysis is performed with three steps for the optimization process.As both the overall estimation error and the local deviation are utilized,the proposed method can achieve a good estimation performance and a relatively uniform estimation error distribution for different feature points.Experimental results on several widely used synthetic sequences and real sequences demonstrate the effectiveness and feasibility of the proposed algorithm.

Study of Array Antenna Pattern Synthesis Based on Sparse Sensing

Aiming at the problem that a large number of array elements are needed for uniform arrays to meet the requirements of direction map,a sparse array pattern synthesis method is proposed in this paper based on the sparse sensing theory.First,the Orthogonal Matching Pursuit(OMP)algorithm and the Exact Augmented Lagrange Multiplier(EALM)algorithm were improved in the sparse sensing theory to obtain a more efficient Orthogonal Multi⁃Matching Pursuit(OMMP)algorithm and the Semi⁃Exact Augmented Lagrange Multiplier(SEALM)algorithm.Then,the two improved algorithms were applied to linear array and planar array pattern syntheses respectively.Results showed that the improved algorithms could achieve the required pattern with very few elements.Numerical simulations verified the effectiveness and superiority of the two synthetic methods.In addition,compared with the existing sparse array synthesis method,the proposed method was more robust and accurate,and could maintain the advantage of easy implementation.

Analysis on a Superlinearly Convergent Augmented Lagrangian Method

The augmented Lagrangian method is a classical method for solving constrained optimization.Recently,the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low rank matrix optimization problems.However,most Lagrangian methods use first order information to update the Lagrange multipliers,which lead to only linear convergence.In this paper,we study an update technique based on second order information and prove that superlinear convergence can be obtained.Theoretical properties of the update formula are given and some implementation issues regarding the new update are also discussed.