A constraint shifting homotopy method for convex multi-objective programming

In this paper, a constraint shifting combined homotopy method for solving multi-objective programming problems with both equality and inequality constraints is presented. It does not need the starting point to be an interior point or a feasible point and hence is convenient to use. Under some assumptions, the existence and convergence of a smooth path to an efficient solution are proven. Simple numerical results are given.     

A constraint shifting homotopy method for convex multi-objective programming

In this paper, a constraint shifting combined homotopy method for solving multi-objective programming problems with both equality and inequality constraints is presented. It does not need the starting point to be an interior point or a feasible point and hence is convenient to use. Under some assumptions, the existence and convergence of a smooth path to an efficient solution are proven. Simple numerical results are given.     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

Nonlinear robust optimization via sequential convex bilevel programming

In this paper, we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min–max problems which arise in the context of semi-infinite programming. Here, our main motivation are nonlinear inequality constrained robust optimization problems. In the first part of the paper, we propose a conservative approximation strategy for such nonlinear and non-convex robust optimization problems: under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given, we show how to formulate a lower-level concave min–max problem which approximates the robust counterpart in a conservative way. This approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties. In the second part of the paper, we review existing theory on optimality conditions for nonlinear lower-level concave min–max problems which arise in the context of semi-infinite programming. Regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem, we end up with a structured mathematical program with complementarity constraints (MPCC). The special hierarchical structure of this MPCC can be exploited in a novel sequential convex bilevel programming algorithm. We discuss the surprisingly strong global and locally quadratic convergence properties of this method, which can in this form neither be obtained with existing SQP methods nor with interior point relaxation techniques for general MPCCs. Finally, we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example.     

一般约束最优化拓广的强次可行方向法

本文讨论非线性等式与不等式最优化问题,引进一个拟罚函数及其相应的只带不等式约束的辅助问题,然后采用广义投影技术和强次可行方向法思想建立原问题的一个全局收敛新算法,该算法具有初点始任意,结构简单,计算量较小等特点。     

A sparse nonlinear optimization algorithm

One of the most effective numerical techniques for solving nonlinear programming problems is the sequential quadratic programming approach. Many large nonlinear programming problems arise naturally in data fitting and when discretization techniques are applied to systems described by ordinary or partial differential equations. Problems of this type are characterized by matrices which are large and sparse. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by these applications. Numerical experience is reported for a collection of trajectory optimization problems with nonlinear equality and inequality constraints.The authors wish to acknowledge the insightful contributions of Dr. William Huffman.     

An elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions in nonlinear programming

In this note we give an elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets.     

ε-Optimal solutions in nondifferentiable convex programming and some related questions

In this paper we present ε-optimality conditions of the Kuhn-Tucker type for points which are within ε of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints, linear equality constraints and abstract constraints. Such ε-optimality conditions are of interest for theoretical consideration as well as from the computational point of view. Some illustrative applications are made. Thus we derive an expression for the ε-subdifferential of a general convex ‘max function’. We also show how the ε-optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints.     

A modified SLP algorithm and its global convergence

This paper concerns a filter technique and its application to the trust region method for nonlinear programming (NLP) problems. We used our filter trust region algorithm to solve NLP problems with equality and inequality constraints, instead of solving NLP problems with just inequality constraints, as was introduced by Fletcher et al. [R. Fletcher, S. Leyffer, Ph.L. Toint, On the global converge of an SLP-filter algorithm, Report NA/183, Department of Mathematics, Dundee University, Dundee, Scotland, 1999]. We incorporate this filter technique into the traditional trust region method such that the new algorithm possesses nonmonotonicity. Unlike the tradition trust region method, our algorithm performs a nonmonotone filter technique to find a new iteration point if a trial step is not accepted. Under mild conditions, we prove that the algorithm is globally convergent.     

一个改进的SQP型算法

本文建立非线性等式和不等式约束规划问题的一个序列二次规划(SQP)型算法.算法的每次迭代只需解一个确实可解的二次规划，然后对其解进行简单的显式校正，便可产生关于罚函数是下降的搜索方向，克服Maratos效应.在适当的假设条件下，还论证了算法的全局收敛性和超级收敛性.     

A CLASS OF REDUCED GRADIENT METHODS FOR HANDLING OPTIMIZATION PROBLEMS WITH LINEAR INEQUALITY CONSTRAINTS

A class of reduced gradient methods for handling general optimization problems with linear equality and inequality constraints is suggested in this paper. Although a slack vector is introduced, the dimension of the problem is not increased, which is unlike the conventional way of transferring the inequality constraints into the equality constraints by introducing slack variables. When an iterate x(k) is not a K-T point of the problem under consideration, different feasible descent directions can be obtained by different choices of the slack vectors. The suggested method is globally convergent and the numerical experiment given in the paper shows that the method is efficient.     

A nonlinear interval number programming method for uncertain optimization problems

In this paper, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method.     

一般广义几何规划问题的一种有效数值方法

一般广义几何规划问题的一种有效数值方法张可村，杨波艇（西安交通大学）ＡＮＥＦＦＥＣＴＩＶＥＮＵＭＥＲＩＣＡＬＭＥＴＨＯＤＦＯＲＧＥＮＥＲＡＬＳＩＧＮＯＭＩＡＬＧＥＯＭＥＴＲＩＣＰＲＯＧＲＡＭＭＩＮＧＰＲＯＢＬＥＭＳ￥ＺｈａｎｇＫｅ－ｃｕｎ；ＹａｎｇＢ...     

A Projected Indefinite Dogleg Path Method for Equality Constrained Optimization

In this paper, we propose a 2-step trust region indefinite dogleg path method for the solution of nonlinear equality constrained optimization problems. The method is a globally convergent modification of the locally convergent Fontecilla method and an indefinite dogleg path method is proposed to get approximate solutions of quadratic programming subproblems even if the Hessian in the model is indefinite. The dogleg paths lie in the null space of the Jacobian matrix of the constraints. An ₁ exact penalty function is used in the method to determine if a trial point is accepted. The global convergence and the local two-step superlinear convergence rate are proved. Some numerical results are presented.     

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.     

Convergence analysis of a nonlinear Lagrangian algorithm for nonlinear programming with inequality constraints

In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an unconstrained programming, convergents locally to a Kuhn-Tucker point of the primal nonlinear programming problem.     

用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

An improved sequential quadratic programming algorithm for solving general nonlinear programming problems

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.     

The asymptotic behavior of nonlinear semigroups and invariant means

Various first-order sufficient optimality criteria for continuous-time nonlinear programming problems with nonlinear equality and inequality constraints are established under generalized convexity assumptions, and applications of these criteria to optimal control and continuous-time fractional programming problems are briefly discussed.     

初始点任意的一个非线性优化的广义梯度投影法

广义投影算法的优点是避免转轴运算。它成功地给出了线性约束问题、初始点任意的只带非线性不等式约束问题，以及利用辅助规划来处理带等式与不等式约束问题的算法．后者完满地解决了投影算法对于非线性等式约束问题的处理，但要求满足不等式约束的初始点．本文据此利用广义投影与罚函数技巧给出了一个初始点任意的等式与不等式约束问题的算法，省去了求初始解的计算，并保持了上述方法的优点，证明了算法的全局收敛性