Topology optimization of space vehicle structures considering attitude control effort

The topology optimization of load-bearing structural components for reducing attitude control efforts of miniature space vehicles is investigated. Based on the derivation of the cold gas consumption rate of three-axis stabilization actuators, it is pointed out that the attitude control efforts associated with cold gas micro thrusters are closely related to the mass moment inertia of the system. Therefore, the need to restrict the mass moments of inertia of the structural components is highlighted in the design of the load-bearing structural components when the attitude control performance is concerned. The optimal layout design of the space vehicle structure considering attitude control effort is, thus, reformulated as a topology optimization problem for minimum compliance under constraints on mass moments of inertia. Numerical techniques for the optimization problem are discussed. For the case of a single constraint on the mass moment of inertia about a given axis, a design variable updating scheme based on the Karush–Kuhn–Tucker optimality criteria is used to solve the minimization problem. For the problem with multiple constraints, mathematical programming approach is employed to seek the optimum. Numerical examples will be given to demonstrate the validity and applicability of the present problem statement.     

具有等式与不等式约束条件拟可微优化的Lagrange乘子型最优性条件

讨论了不等式约束优化问题中拟微分形式下Ｆｒｉｔｚ Ｊｏｈｎ必要条件与 Ｃｌａｒｋｅ广义梯度形式下Ｆｒｉｔｚ Ｊｏｈｎ必要条件的关系．在较弱条件下给出了具有等式与不等式约束条件的两个Ｌａｇｒａｎｇｅ乘子形式的最优性必要条件,在这两个条件中等式约束函数的拟微分和Ｃｌａｒｋｅ广义梯度分别被使用。     

Nonlinear programming without differentiability in banach spaces: necessary and sufficient constraint qualification

The problem under consideration is a maximization problem over a constraint set defined by a finite number of inequality and equality constraints over an arbitrary set in a reflexive Banach space. A generalization of the Kuhn-Tucker necessary conditions is developed where neither the objective function nor the constraint functions are required to be differentiable. A new constraint qualification is imposed in order to validate the optimality criteria. It is shown that this qualification is the weakest possible in the sense that it is necessary for the optimality criteria to hold at the point under investigation for all families of objective functions having a constrained local maximum at this point     

On the application of Newton-type methods to Fritz John optimality conditions

An approach to the numerical solution of optimization problems with equality constraints violating the traditional constraint qualification is developed. According to this approach, an (overdetermined) defining system is constructed based on the Fritz John optimality conditions and the Gauss-Newton method is applied to this system. The assumptions required for the implementability and local superlinear convergence of the resulting algorithm are completely characterized in terms of the original problem.     

基于变体积约束的阻尼材料微结构拓扑优化研究北大核心CSCD

阻尼复合结构的抑振性能取决于材料布局和阻尼材料特性.该文提出了一种变体积约束的阻尼材料微结构拓扑优化方法,旨在以最小的材料用量获得具有期望性能的阻尼材料微结构.基于均匀化方法,建立阻尼材料三维微结构有限元模型,得到阻尼材料的等效弹性矩阵.逆用Hashin-Shtrikman界限理论,估计对应于期望等效模量的阻尼材料体积分数限,并构建阻尼材料体积约束限的移动准则.将获得阻尼材料微结构期望性能的优化问题转化为体积约束下最大化等效模量的优化问题,建立阻尼材料微结构的拓扑优化模型.利用优化准则法更新设计变量,实现最小材料用量下的阻尼材料微结构最优拓扑设计.通过典型数值算例验证了该方法的可行性和有效性,并讨论了初始微构型、网格依赖性和弹性模量等对阻尼材料微结构的影响.     

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.     

The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems

This paper considers the problem of minimizing a continuously differentiable function with a Lipschitz continuous gradient subject to a single linear equality constraint and additional bound constraints on the decision variables. We introduce and analyze several variants of a 2-coordinate descent method: a block descent method that performs an optimization step with respect to only two variables at each iteration. Based on two new optimality measures, we establish convergence to stationarity points for general nonconvex objective functions. In the convex case, when all the variables are lower bounded but not upper bounded, we show that the sequence of function values converges at a sublinear rate. Several illustrative numerical examples demonstrate the effectiveness of the method.     

A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints

In this paper, we consider an optimal zero-forcing beamformer design problem in multi-user multiple-input multiple-output broadcast channel. The minimum user rate is maximized subject to zero-forcing constraints and power constraint on each base station antenna array element. The natural formulation leads to a nonconvex optimization problem. This problem is shown to be equivalent to a convex optimization problem with linear objective function, linear equality and inequality constraints and quadratic inequality constraints. Here, the indirect elimination method is applied to reduce the convex optimization problem into an equivalent convex optimization problem of lower dimension with only inequality constraints. The primal-dual interior point method is utilized to develop an effective algorithm (in terms of computational efficiency) via solving the modified KKT equations with Newton method. Numerical simulations are carried out. Compared to algorithms based on a trust region interior point method and sequential quadratic programming method, it is observed that the method proposed is much superior in terms of computational efficiency.     

Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.     

Exact penalties for optimization problems with 2-regular equality constraints

A new first-order sufficient condition for penalty exactness that includes neither the standard constraint qualification requirement nor the second-order sufficient optimality condition is proposed for optimization problems with equality constraints.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Strong Kuhn–Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems

We consider a Pareto multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and inequality constraints are locally Lipschitz, and the equality constraints are Fréchet differentiable. We study several constraint qualifications in the line of Maeda (J. Optim. Theory Appl. 80: 483–500, 1994) and, under the weakest ones, we establish strong Kuhn–Tucker necessary optimality conditions in terms of Clarke subdifferentials so that the multipliers of the objective functions are all positive.     

Handling equality constraints in evolutionary optimization

Over the last few decades several methods have been proposed for handling functional constraints while solving optimization problems using evolutionary algorithms (EAs). However, the presence of equality constraints makes the feasible space very small compared to the entire search space. As a consequence, the handling of equality constraints has long been a difficult issue for evolutionary optimization methods. This paper presents a Hybrid Evolutionary Algorithm (HEA) for solving optimization problems with both equality and inequality constraints. In HEA, we propose a new local search technique with special emphasis on equality constraints. The basic concept of the new technique is to reach a point on the equality constraint from the current position of an individual solution, and then explore on the constraint landscape. We believe this new concept will influence the future research direction for constrained optimization using population based algorithms. The proposed algorithm is tested on a set of standard benchmark problems. The results show that the proposed technique works very well on those benchmark problems.     

ε-Pareto Optimality Conditions for Convex Multiobjective Programming via Max Function

We consider a nondifferentiable convex multiobjective optimization problem whose feasible set is defined by affine equality constraints, convex inequality constraints, and an abstract convex set constraint. We obtain Fritz John and Kuhn–Tucker necessary and sufficient conditions for ε-Pareto optimality via a max function. We also provide some relations among ε-Pareto solutions for such a problem and approximate solutions for several associated scalar problems.     

An interior affine scaling projective algorithm for nonlinear equality and linear inequality constrained optimization

In this paper, we propose a new nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving optimization subject to nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the nonlinear equality and linear inequality constrained optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint in a null subspace of the tangential space, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. By introducing the Fletcher's penalty function as the merit function, trust region strategy with interior point backtracking technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. The global convergence of the proposed algorithm while maintaining fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some high nonlinear function conditioned cases.     

LMI based design of constrained fuzzy predictive control

Predictive control of nonlinear systems subject to output and input constraints is considered. A fuzzy model is used to predict the future behavior. Two new ideas are proposed here. First, an added constraint on the applied control action is used to ensure the decrease of a quadratic Lyapunov function, and so guarantee Lyapunov exponential stability of the closed-loop system. Second, the feasibility of the finite-horizon optimization problem with the added constraints is ensured based on an off-line solution of a set of LMIs. The novel stability method is compared to the existing methods, such as the techniques based on the end-point constraints (terminal constraint set), and the robust stability techniques based on the small gain theory. The proposed method ensures Lyapunov exponential stability, does not need an auxiliary controller and can be used with any feasible controller parameters. Illustrative examples including the predictive control of a highly nonlinear chemical reactor (CSTR) are discussed.     

An efficient curvilinear method for the minimization of a nonlinear function subject to linear inequality constraints

An algorithm is presented that minimizes a continuously differentiable function in several variables subject to linear inequality constraints. At each step of the algorithm an arc is generated along which a move is performed until either a point yielding a sufficient descent in the function value is determined or a constraint boundary is encountered. The decision to delite a constraint from the list of active constraints is based upon periodic estimates of the Kuhn-Tucker multipliers. The curvilinear search paths are obtained by solving a linear approximation to the differential equation of the continuous steepest descent curve for the objective function on the equality constrained region defined by the constraints which are required to remain binding. If the Hessian matrix of the objective function has certain properties and if the constraint gradients are linearly independent, the sequence generated by the algorithm converges to a point satisfying the Kuhn-Tucker optimality conditions at a rate that is at least quadratic.     

Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.     

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.