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.

Capacity Optimization Using Augmented Lagrange Method in Intelligent Reflecting Surface-Based MIMO Communication Systems

In this paper,we present an algorithm for capacity optimization in intelligent reflecting surface(IRS)-based multiple-input multiple-output(MIMO)communication systems.To maximize the capacity of elements in IRS,we use augmented Lagrange method with the equivalent transformations on the covariance matrix and reflection matrix constraints.This results an adjustable phase shift on the incident signal.Furthermore,we reshape the complex-valued covariance matrix and reflection matrix to a vector for the ease of calculating partial derivatives to find the search direction.Then,the quasi-Newton updates and modified Broyden-Fletcher-Goldfarb-Shano(BFGS)method in the complex domain form are used to find the local minimum.Finally,numerical simulation results demonstrate that our proposed IRS-aided system using the algorithm performs better than the state-of-the-art and the conventional communication systems.

Reduced projection augmented Lagrange bi-conjugate gradient method for contact and impact problems

Based on the numerical governing formulation and non-linear complementary conditions of contact and impact problems, a reduced projection augmented Lagrange bi- conjugate gradient method is proposed for contact and impact problems by translating non-linear complementary conditions into equivalent formulation of non-linear program- ming. For contact-impact problems, a larger time-step can be adopted arriving at numer- ical convergence compared with penalty method. By establishment of the impact-contact formulations which are equivalent with original non-linear complementary conditions, a reduced projection augmented Lagrange bi-conjugate gradient method is deduced to im- prove precision and efficiency of numerical solutions. A numerical example shows that the algorithm we suggested is valid and exact.

A New Algorithm for Reducing Dimensionality of L1-CSVM Use Augmented Lagrange Method

Principal component analysis and generalized low rank approximation of matrices are two different dimensionality reduction methods. Two different dimensionality reduction algorithms are applied to the L1-CSVM model based on augmented Lagrange method to explore the variation of running time and accuracy of the model in dimensionality reduction space. The results show that the improved algorithm can greatly reduce the running time and improve the accuracy of the algorithm.

Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Lowner operator associated with a potential function for the optimization problems with inequality constraints.The favorable properties of both the Lowner operator and the corresponding augmented Lagrangian are discussed.And under some mild assumptions,the rate of convergence of the augmented Lagrange algorithm is studied in detail.

Structure Analysis of Locking Mechanism of Gear-Rack Typed Ship-Lift

Contact nonlinear theory was researched. Contact problem was transformed into optimization problem containing Lagrange multiplier, and unsymmetrical stiffness matrix was transformed into symmetrical stiffness matrix. A finite element analysis （FEA） model defining more than 300 contact pairs for long nut-short screw locking mechanism of a large-scale vertical gear-rack typed ship-lift was built. Using augmented Lagrange method and symmetry algorithm of contact element stiffness, the FEA model was analyzed, and the contact stress of contact interfaces and the von Mises stress of key parts were obtained. The results show that the design of the locking mechanism meets the requirement of engineering, and this method is effective for solving large stole nonlinear contact pairs.

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.

Optimization of multibody systems based on the generalized-α projection method for DAEs

Efficient optimization strategy of multibody systems is developed in this paper. Aug- mented Lagrange method is used to transform constrained optimal problem into unconstrained form firstly. Then methods based on second order sensitivity are used to solve the unconstrained problem, where the sensitivity is solved by hybrid method. Generalized-α method and generalized-α projection method for the differential-algebraic equation, which shows more efficient properties with the lager time step, are presented to get state variables and adjoint variables during the optimization procedure. Numerical results validate the accuracy and efficiency of the methods is presented.

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.

A NEW TRUST-REGION ALGORITHM FOR NONLINEAR CONSTRAINED OPTIMIZATION

We propose a new trust region algorithm for nonlinear constrained optimization problems. In each iteration of our algorithm, the trial step is computed by minimizing a quadratic approximation to the augmented Lagrange function in the trust region. The augmented Lagrange function is also used as a merit function to decide whether the trial step should be accepted. Our method extends the traditional trust region approach by combining a filter technique into the rules for accepting trial steps so that a trial step could still be accepted even when it is rejected by the traditional rule based on merit function reduction. An estimate of the Lagrange multiplier is updated at each iteration, and the penalty parameter is updated to force sufficient reduction in the norm of the constraint violations. Active set technique is used to handle the inequality constraints. Numerical results for a set of constrained problems from the CUTEr collection are also reported.

A Continuation Algorithm for Max-Cut Problem

A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.