Sequential quadratically constrained quadratic programming norm-relaxed algorithm of strongly sub-feasible directions

In this paper, we present a sequential quadratically constrained quadratic programming (SQCQP) norm-relaxed algorithm of strongly sub-feasible directions for the solution of inequality constrained optimization problems. By introducing a new unified line search and making use of the idea of strongly sub-feasible direction method, the proposed algorithm can well combine the phase of finding a feasible point (by finite iterations) and the phase of a feasible descent norm-relaxed SQCQP algorithm. Moreover, the former phase can preserve the “sub-feasibility” of the current iteration, and control the increase of the objective function. At each iteration, only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction. Without any other correctional directions, the global, superlinear and a certain quadratic convergence (which is between 1-step and 2-step quadratic convergence) properties are proved under reasonable assumptions. Finally, some preliminary numerical results show that the proposed algorithm is also encouraging.     

A sequential quadratically constrained quadratic programming method with an augmented Lagrangian line search function

Based on an augmented Lagrangian line search function, a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems. Compared to quadratic programming solved in the traditional SQP methods, a convex quadratically constrained quadratic programming is solved here to obtain a search direction, and the Maratos effect does not occur without any other corrections. The “active set” strategy used in this subproblem can avoid recalculating the unnecessary gradients and (approximate) Hessian matrices of the constraints. Under certain assumptions, the proposed method is proved to be globally, superlinearly, and quadratically convergent. As an extension, general problems with inequality and equality constraints as well as nonmonotone line search are also considered.     

A Computational Study of the Homogeneous Algorithm for Large-scale Convex Optimization

Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method; nevertheless, it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems, which also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient.     

Global optimization of a class of nonconvex quadratically constrained quadratic programming problems

In this paper we study a class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.     

A constrained optimization based method for acoustic finite element model updating of cavities using pressure response

A method based on constrained optimization for updating of an acoustic finite element model using pressure response is proposed in this paper. The constrained optimization problem is solved using sequential quadratic programming algorithm. Updating parameters related to the properties of the sound absorbers and the measurement errors are considered. Effectiveness of the method is demonstrated by numerical studies on a 2D rectangular cavity and a car cavity. It is shown that the constrained formulation, that includes lower and upper bounds on the updating parameters in the form of inequality constraints, is important for obtaining a correct updated model. It is seen that the proposed updating method is not only able to effectively update the model to obtain a close match between the finite element model pressure response and the reference pressure response, but is also able to identify the correction factors to the parameters in error with reasonable accuracy.     

A working set SQCQP algorithm with simple nonmonotone penalty parameters

In this paper, we present a new sequential quadratically constrained quadratic programming (SQCQP) algorithm, in which a simple updating strategy of the penalty parameter is adopted. This strategy generates nonmonotone penalty parameters at early iterations and only uses the multiplier corresponding to the bound constraint of the quadratically constrained quadratic programming (QCQP) subproblem instead of the multipliers of the quadratic constraints, which will bring some numerical advantages. Furthermore, by using the working set technique, we remove the constraints of the QCQP subproblem that are locally irrelevant, and thus the computational cost could be reduced. Without assuming the convexity of the objective function or the constraints, the algorithm is proved to be globally, superlinearly and quadratically convergent. Preliminary numerical results show that the proposed algorithm is very promising when compared with the tested SQP algorithms.     

A working set SQCQP algorithm with simple nonmonotone penalty parameters

In this paper, we present a new sequential quadratically constrained quadratic programming (SQCQP) algorithm, in which a simple updating strategy of the penalty parameter is adopted. This strategy generates nonmonotone penalty parameters at early iterations and only uses the multiplier corresponding to the bound constraint of the quadratically constrained quadratic programming (QCQP) subproblem instead of the multipliers of the quadratic constraints, which will bring some numerical advantages. Furthermore, by using the working set technique, we remove the constraints of the QCQP subproblem that are locally irrelevant, and thus the computational cost could be reduced. Without assuming the convexity of the objective function or the constraints, the algorithm is proved to be globally, superlinearly and quadratically convergent. Preliminary numerical results show that the proposed algorithm is very promising when compared with the tested SQP algorithms.     

Branch and bound algorithm for multidimensional scaling with city-block metric

A two level global optimization algorithm for multidimensional scaling (MDS) with city-block metric is proposed. The piecewise quadratic structure of the objective function is employed. At the upper level a combinatorial global optimization problem is solved by means of branch and bound method, where an objective function is defined as the minimum of a quadratic programming problem. The later is solved at the lower level by a standard quadratic programming algorithm. The proposed algorithm has been applied for auxiliary and practical problems whose global optimization counterpart was of dimensionality up to 24.     

A Sequential Quadratically Constrained Quadratic Programming Method of Feasible Directions

In this paper, a sequential quadratically constrained quadratic programming method of feasible directions is proposed for the optimization problems with nonlinear inequality constraints. At each iteration of the proposed algorithm, a feasible direction of descent is obtained by solving only one subproblem which consist of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the functions of the discussed problems, and such a subproblem can be formulated as a second-order cone programming which can be solved by interior point methods. To overcome the Maratos effect, an efficient higher-order correction direction is obtained by only one explicit computation formula. The algorithm is proved to be globally convergent and superlinearly convergent under some mild conditions without the strict complementarity. Finally, some preliminary numerical results are reported. Project supported by the National Natural Science Foundation (No. 10261001), Guangxi Science Foundation (Nos. 0236001, 064001), and Guangxi University Key Program for Science and Technology Research (No. 2005ZD02) of China.     

Solving nonlinear programming problems with very many constraints

For solving the smooth constrained nonlinear programming problem, sequential quadratic programming (SQP) methods are considered to be the standard tool, as long as they are applicable. However one possible situation preventing the successful solution by a standard SQP-technique, arises if problems with a very large number of constraints are to be solved. Typical applications are semi-infinite or min-max optimization, optimal control or mechanical structural optimization. The proposed technique proceeds from a user defined number of linearized constraints, that is to be used internally to determine the size of the quadratic programming subproblem. Significant constraints are then selected automatically by the algorithm. Details of the numerical implementation and some experimental results are presented     

Advances in concrete arch dams shape optimization

This paper presents an efficient methodology to find the optimum shape of arch dams. In order to create the geometry of arch dams a new algorithm based on Hermit Splines is proposed. A finite element based shape sensitivity analysis for design-dependent loadings involving body force, hydrostatic pressure and earthquake loadings is implemented. The sensitivity analysis is performed using the concept of mesh design velocity. In order to consider the practical requirements in the optimization model such as construction stages, many geometrical and behavioral constrains are included in the model in comparison with previous researches. The optimization problem is solved via the sequential quadratic programming (SQP) method. The proposed methods are applied successfully to an Iranian arch dam, and good results are achieved. By using such methodology, efficient software for shape optimization of concrete arch dams for practical and reliable design now is available.     

Simple Sequential Quadratically Constrained Quadratic Programming Feasible Algorithm with Active Identification Sets for Constrained Minimax Problems

In this paper, the nonlinear minimax problems with inequality constraints are discussed. Based on the idea of simple sequential quadratically constrained quadratic programming algorithm for smooth constrained optimization, an alternative algorithm for solving the discussed problems is proposed. Unlike the previous work, at each iteration, a feasible direction of descent called main search direction is obtained by solving only one subprogram which is composed of a convex quadratic objective function and simple quadratic inequality constraints without the second derivatives of the constrained functions. Then a high-order correction direction used to avoid the Maratos effect is computed by updating the main search direction with a system of linear equations. The proposed algorithm possesses global convergence under weak Mangasarian–Fromovitz constraint qualification and superlinear convergence under suitable conditions with the upper-level strict complementarity. At last, some preliminary numerical results are reported.     

Least squares problems with inequality constraints as quadratic constraints

Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and one from a hydrological application. Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ² regularization method. The χ² regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.     

An Extended Sequential Quadratically Constrained Quadratic Programming Algorithm for Nonlinear, Semidefinite, and Second-Order Cone Programming

This paper is concerned with nonlinear, semidefinite, and second-order cone programs. A general algorithm, which includes sequential quadratic programming and sequential quadratically constrained quadratic programming methods, is presented for solving these problems. In the particular case of standard nonlinear programs, the algorithm can be interpreted as a prox-regularization of the Solodov sequential quadratically constrained quadratic programming method presented in Mathematics of Operations Research (2004). For such type of methods, the main cost of computation amounts to solve a linear cone program for which efficient solvers are available. Usually, “global convergence results” for these methods require, as for the Solodov method, the boundedness of the primal sequence generated by the algorithm. The other purpose of this paper is to establish global convergence results without boundedness assumptions on any of the iterative sequences built by the algorithm.     

混合互补问题牛顿型算法的二阶收敛性

在凸规划理论中,通过ＫＴ条件,往往将约束最优化问题归结为一个混合互补问题来求解。该文就正则解和一般解两种情形分别给出了求解混合互补问题牛顿型算法的二阶收敛性的充分性条件,并在一定条件下证明了非精确牛顿法和离散牛顿法所具有的二阶收敛性。     

A very simple SQCQP method for a class of smooth convex constrained minimization problems with nice convergence results

We introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods, called sequential simple quadratic method (SSQM). The computational simplicity of SSQM, which uses first-order information, makes it suitable for large scale problems. Theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations. When in addition the objective function is strongly convex then asymptotic linear rate of convergence is established.     

A branch and cut algorithm for nonconvex quadratically constrained quadratic programming

We present a branch and cut algorithm that yields in finite time, a globally ε-optimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported that include standard test problems taken from the literature. Some of these problems are solved for the first time with a proof of global optimality. Received December 19, 1997 / Revised version received July 26, 1999?Published online November 9, 1999     

A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface

By combining FETI algorithms of dual-primal type with recent results for bound constrained quadratic programming problems, we develop an optimal algorithm for the numerical solution of coercive variational inequalities. The model problem is discretized using non-penetration conditions of mortar type across the potential contact interface, and a FETI-DP algorithm is formulated. The resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem. Numerical experiments for non-matching meshes across the contact interface confirm the theoretical scalability of the algorithm.     

边界约束非凸二次规划问题的分枝定界方法

本文是研究带有边界约束非凸二次规划问题,我们把球约束二次规划问题和线性约束凸二次规划问题作为子问题,分明引用了它们的一个求整体最优解的有效算法,我们提出几种定界的紧、松驰策略,给出了求解原问题整体最优解的分枝定界算法,并证明了该算法的收敛性,不同的定界组合就可以产生不同的分枝定界算法,最后我们简单讨论了一般有界凸域上非凸二次规划问题求整体最优解的分枝与定界思想。     

一种求解二次约束二次规划问题的自适应全局优化算法

为了更好地解决二次约束二次规划问题(QCQP), 本文基于分支定界算法框架提出了自适应线性松弛技术, 在理论上证明了这种新的定界技术对于解决(QCQP)是可观的。文中分支操作采用条件二分法便于对矩形进行有效剖分; 通过缩减技术删除不包含全局最优解的部分区域, 以加快算法的收敛速度。最后, 通过数值结果表明提出的算法是有效可行的。