Multilevel Design Optimization Under Uncertainty with Application to Product-Material Systems

The main objective of this research is to develop a computational design tool for multilevel optimization of product-material systems under uncertainty. To accomplish this goal, an exponential penalty function (EPF) formulation based on method of multipliers is developed for solving multilevel optimization problems within the framework of Analytical Target Cascading (ATC). The original all-at-once constrained optimization problem is decomposed into a hierarchical system with consistency constraints enforcing the target-response coupling in the connected elements. The objective function is combined with the consistency constraints in each element to formulate an augmented Lagrangian with EPF. The EPF formulation is implemented using double-loop (EPF I) and single-loop (EPF II) coordination strategies and two penalty-parameter-updating schemes. The computational characteristics of the proposed approaches are investigated using different nonlinear convex and non-convex optimization problems. An efficient reliability-based design optimization method, Single Loop Single Vector (SLSV), is integrated with Augmented Lagrangian (AL) formulation of ATC for solution of hierarchical multilevel optimization problems under uncertainty. In the proposed SLSV+AL approach, the uncertainties are propagated by matching the required moments of connecting responses/targets and linking variables present in the decomposed system. The accuracy and computational efficiency of SLSV+AL are demonstrated through the solution of different benchmark problems and comparison of results with those from other optimization methods. Finally, the developed computational design optimization tool is used for design optimization of hybrid multiscale composite sandwich plates with/without uncertainty. Both carbon nanofiber (CNF) waviness and CNF-matrix interphase properties are included in the model. By decomposing the sandwich plate, structural and material designs are combined and treated as a multilevel optimization problem. The application problem considers the minimum-weight design of an in-plane loaded sandwich plate with a honeycomb core and laminated composite face sheets that are reinforced by both conventional continuous fibers and CNF-enhanced polymer matrix. Besides global buckling, shear crimping, intracell buckling, and face sheet wrinkling are also treated as design constraints

Multilevel Design Optimization Under Uncertainty with Application to Product-Material Systems

The main objective of this research is to develop a computational design tool for multilevel optimization of product-material systems under uncertainty. To accomplish this goal, an exponential penalty function (EPF) formulation based on method of multipliers is developed for solving multilevel optimization problems within the framework of Analytical Target Cascading (ATC). The original all-at-once constrained optimization problem is decomposed into a hierarchical system with consistency constraints enforcing the target-response coupling in the connected elements. The objective function is combined with the consistency constraints in each element to formulate an augmented Lagrangian with EPF. The EPF formulation is implemented using double-loop (EPF I) and single-loop (EPF II) coordination strategies and two penalty-parameter-updating schemes. The computational characteristics of the proposed approaches are investigated using different nonlinear convex and non-convex optimization problems. An efficient reliability-based design optimization method, Single Loop Single Vector (SLSV), is integrated with Augmented Lagrangian (AL) formulation of ATC for solution of hierarchical multilevel optimization problems under uncertainty. In the proposed SLSV+AL approach, the uncertainties are propagated by matching the required moments of connecting responses/targets and linking variables present in the decomposed system. The accuracy and computational efficiency of SLSV+AL are demonstrated through the solution of different benchmark problems and comparison of results with those from other optimization methods. Finally, the developed computational design optimization tool is used for design optimization of hybrid multiscale composite sandwich plates with/without uncertainty. Both carbon nanofiber (CNF) waviness and CNF-matrix interphase properties are included in the model. By decomposing the sandwich plate, structural and material designs are combined and treated as a multilevel optimization problem. The application problem considers the minimum-weight design of an in-plane loaded sandwich plate with a honeycomb core and laminated composite face sheets that are reinforced by both conventional continuous fibers and CNF-enhanced polymer matrix. Besides global buckling, shear crimping, intracell buckling, and face sheet wrinkling are also treated as design constraints

Optimal mistuning for enhanced aeroelastic stability of transonic fans

An inverse design procedure was developed for the design of a mistuned rotor. The design requirements are that the stability margin of the eigenvalues of the aeroelastic system be greater than or equal to some minimum stability margin, and that the mass added to each blade be positive. The objective was to achieve these requirements with a minimal amount of mistuning. Hence, the problem was posed as a constrained optimization problem. The constrained minimization problem was solved by the technique of mathematical programming via augmented Lagrangians. The unconstrained minimization phase of this technique was solved by the variable metric method. The bladed disk was modelled as being composed of a rigid disk mounted on a rigid shaft. Each of the blades were modelled with a single tosional degree of freedom

An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming

Powerful interior-point methods (IPM) based commercial solvers, such as Gurobi and Mosek, have been hugely successful in solving large-scale linear programming (LP) problems. The high efficiency of these solvers depends critically on the sparsity of the problem data and advanced matrix factorization techniques. For a large scale LP problem with data matrix $A$ that is dense (possibly structured) or whose corresponding normal matrix $AA^T$ has a dense Cholesky factor (even with re-ordering), these solvers may require excessive computational cost and/or extremely heavy memory usage in each interior-point iteration. Unfortunately, the natural remedy, i.e., the use of iterative methods based IPM solvers, although can avoid the explicit computation of the coefficient matrix and its factorization, is not practically viable due to the inherent extreme ill-conditioning of the large scale normal equation arising in each interior-point iteration. To provide a better alternative choice for solving large scale LPs with dense data or requiring expensive factorization of its normal equation, we propose a semismooth Newton based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can efficiently be used to solve simpler yet better conditioned semismooth Newton linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic superlinear convergence but is also proven to enjoy a finite termination property. Numerical comparisons with Gurobi have demonstrated encouraging potential of {\sc Snipal} for handling large-scale LP problems where the constraint matrix $A$ has a dense representation or $AA^T$ has a dense factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil

A Partially Feasible Distributed SQO Method for Two-block General Linearly Constrained Smooth Optimization

This paper discusses a class of two-block smooth large-scale optimization problems with both linear equality and linear inequality constraints, which have a wide range of applications, such as economic power dispatch, data mining, signal processing, etc.Our goal is to develop a novel partially feasible distributed (PFD) sequential quadratic optimization (SQO) method (PFD-SQO method) for this kind of problems. The design of the method is based on the ideas of SQO method and augmented Lagrangian Jacobian splitting scheme as well as feasible direction method,which decomposes the quadratic optimization (QO) subproblem into two small-scale QOs that can be solved independently and parallelly. A novel disturbance contraction term that can be suitably adjusted is introduced into the inequality constraints so that the feasible step size along the search direction can be increased to 1. The new iteration points are generated by the Armijo line search and the partially augmented Lagrangian function that only contains equality constraints as the merit function. The iteration points always satisfy all the inequality constraints of the problem. The theoretical properties, such as global convergence, iterative complexity, superlinear and quadratic rates of convergence of the proposed PFD-SQO method are analyzed under appropriate assumptions, respectively. Finally, the numerical effectiveness of the method is tested on a class of academic examples and an economic power dispatch problem, which shows that the proposed method is quite promising