Interval Analysis on Directed Acyclic Graphs for Global Optimization

A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the flow of the computation. Using bounds on ranges of intermediate results, represented as weights on the nodes and a suitable mix of forward and backward evaluation, it is possible to give efficient implementations of interval evaluation and automatic differentiation. It is shown how to combine this with constraint propagation techniques to produce narrower interval derivatives and slopes than those provided by using only interval automatic differentiation preceded by constraint propagation. The implementation is based on earlier work by L.V. Kolev, (1997), Reliable Comput., 3, 83–93 on optimal slopes and by C. Bliek, (1992), Computer Methods for Design Automation, PhD Thesis, Department of Ocean Engineering, Massachusetts Institute of Technology on backward slope evaluation. Care is taken to ensure that rounding errors are treated correctly. Interval techniques are presented for computing from the DAG useful redundant constraints, in particular linear underestimators for the objective function, a constraint, or a Lagrangian. The linear underestimators can be found either by slope computations, or by recursive backward underestimation. For sufficiently sparse problems the work is proportional to the number of operations in the calculation of the objective function (resp. the Lagrangian). Mathematics Subject Classification (2000). primary 65G40, secondary 90C26     

Determination of the composition of the insurance and investment portfolios of a casualty insurance company

This paper presents a model in which expected return is maximized subject to two chance constraints. The first chance constraint requires that the total return is greater than some lower bound with a stipulated probability. The second chance constraint is used to maintain the firm's liquidity. An evaluation of the solution is given. The influence of a premium change on the data requirements of the model is discussed.     

On constraint qualifications

The linear independence constraint qualification (LICQ) and the weaker Mangasarian-Fromovitz constraint qualification (MFCQ) are well-known concepts in nonlinear optimization. A theorem is proved suggesting that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense. As an auxiliary result, it is shown that, under MFCQ, the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.The author wishes to thank Professor H. T. Jongen for valuable advice.     

Optimality conditions and constraint qualifications in Banach space

In this paper, necessary optimality conditions for nonlinear programs in Banach spaces and constraint qualifications for their applicability are considered. A new optimality condition is introduced, and a constraint qualification ensuring the validity of this condition is given. When the domain space is a reflexive space, it is shown that the qualification is the weakest possible. If a certain convexity assumption is made, then this optimality condition is shown to reduce to the well-known extension of the Kuhn-Tucker conditions to Banach spaces. In this case, the constraint qualification is weaker than those previously given.This work was supported in part by the Office of Naval Research, Contract Number N00014-67-A-0321-0003 (NRO 47-095).     

基于时间约束的生产过程优化问题研究

本文提出一个实际的生产过程优化问题:基于时间约束的生产过程优化问题。客户要求企业在规定时间内完成指定批量工件的生产任务,该问题便是从中引出的。该问题的目标是在满足生产时间的条件下最小化总生产成本。本文为该问题建立了整数规划模型。然后以某厂工作缸生产过程为例,采用数学规划软件Cplex 9.0求解模型。     

非凸优化问题的一个对偶结论

本文研究了单约束条件的非凸极小问题的对偶形式,我们的结论是通过变换,可以化成无缝对偶情形,同时我们研究了多约束条件的同类问题的处理方法。     

Search Heuristics for Box Decomposition Methods

In this paper we study search heuristics for box decomposition methods that solve problems such as global optimization, minimax optimization, or quantified constraint solving. For this we unify these methods under a branch-and-bound framework, and develop a model that is more convenient for studying heuristics for such algorithms than the traditional models from Artificial Intelligence. We use the result to prove various theorems about heuristics and apply the outcome to the box decomposition methods under consideration. We support the findings with timings for the method of quantified constraint solving developed by the author.     

Global convergence of a robust filter SQP algorithm

We present a robust filter SQP algorithm for solving constrained optimization problems. This algorithm is based on the modified quadratic programming proposed by Burke to avoid the infeasibility of the quadratic programming subproblem at each iteration. Compared with other filter SQP algorithms, our algorithm does not require any restoration phase procedure which may spend a large amount of computation. The main advantage of our algorithm is that it is globally convergent without requiring strong constraint qualifications, such as Mangasarian–Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ). Furthermore, the feasible limit points of the sequence generated by our algorithm are proven to be the KKT points if some weaker conditions are satisfied. Numerical results are also presented to show the efficiency of the algorithm.     

A General Approach for Constraint Solving by Local Search

In this paper, we present a general approach for solving constraint problems by local search. The proposed approach is based on a set of high-level constraint primitives motivated by constraint programming systems. These constraints constitute the basic bricks to formulate a given combinatorial problem. A tabu search engine ensures the resolution of the problem so formulated. Experimental results are shown to validate the proposed approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

Restricted linear constrained minimization of quadratic functionals

In this work, a linearly constrained minimization of a positive semidefinite quadratic functional is examined. We propose two different approaches to this problem. Our results are concerning infinite dimensional real Hilbert spaces, with a singular positive semidefinite operator related to the functional, and considering as constraint a singular operator. The difference between the proposed approaches for the minimization and previous work on this problem is that it is considered for all vectors belonging to the least squares solutions set, or to the vectors perpendicular to the kernel of the related operator or matrix.     

Multiobjective Search Algorithm with Subdivision Technique

This paper presents a multiobjective search algorithm with subdivision technique (MOSAST) for the global solution of multiobjective constrained optimization problems with possibly noncontinuous objective or constraint functions. This method is based on a random search method and a new version of the Graef-Younes algorithm and it uses a subdivision technique. Numerical results are given for bicriterial test problems.     

Computational Discretization Algorithms for Functional Inequality Constrained Optimization

In this paper, a functional inequality constrained optimization problem is studied using a discretization method and an adaptive scheme. The problem is discretized by partitioning the interval of the independent parameter. Two methods are investigated as to how to treat the discretized optimization problem. The discretization problem is firstly converted into an optimization problem with a single nonsmooth equality constraint. Since the obtained equality constraint is nonsmooth and does not satisfy the usual constraint qualification condition, relaxation and smoothing techniques are used to approximate the equality constraint via a smooth inequality constraint. This leads to a sequence of approximate smooth optimization problems with one constraint. An adaptive scheme is incorporated into the method to facilitate the computation of the sum in the inequality constraint. The second method is to apply an adaptive scheme directly to the discretization problem. Thus a sequence of optimization problems with a small number of inequality constraints are obtained. Convergence analysis for both methods is established. Numerical examples show that each of the two proposed methods has its own advantages and disadvantages over the other.     

求解一类逆鲁棒优化问题的牛顿型扰动方法

逆优化问题是指通过调整目标函数和约束中的某些参数使得已知的一个解成为参数调整后的优化问题的最优解.本文考虑求解一类逆鲁棒优化问题.首先,我们将该问题转化为带有一个线性等式约束,一个二阶锥互补约束和一个线性互补约束的极小化问题;其次,通过一类扰动方法来对转化后的极小化问题进行求解,然后利用带Armijo线搜索的非精确牛顿法求解每一个扰动问题.最后,通过数值实验验证该方法行之有效.     

Comment to “interactive fuzzy goal programming approach for bilevel programming problem” by S.R. Arora and R. Gupta

The fuzzy bilevel programming problem is solved in the paper [1] using an approach of fuzzy goal programming. Using a simple example we show that this will not lead to a satisfactory solution. Especially the fuzzy constraint that the upper level variable is near some desired one has no influence on the computed solution in the example. This constraint is used to model the hierarchy in the approach in [1]. At the end of the paper we suggest one straightforward possibility for modeling the fuzzy bilevel programming problem and converting it into a crisp substitute.     

Optimal Sequential Sampling Policy of Partitioned Random Search and Its Approximation

The paper studies the optimal sequential sampling policy of the partitioned random search (PRS) and its approximation. The PRS is a recently proposed approach for function optimization. It takes explicitly into consideration computation time or cost, assuming that there exist both a cost for each function evaluation and a finite total computation time constraint. It is also motivated at improving efficiency of the widely used crude random search. In particular, the PRS considers partitioning the search region of an objective function into K subregions and employing an independent and identically distributed random sampling scheme for each of K subregions. A sampling policy decides when to terminate the sampling process or which subregion to be sampled next.     

On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case

Some versions of constraint qualifications in the semidifferentiable case are considered for a multiobjective optimization problem with inequality constraints. A Maeda-type constraint qualification is given and Kuhn–Tucker-type necessary conditions for efficiency are obtained. In addition, some conditions that ensure the Maeda-type constraint qualification are stated.     

Minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint

In this paper, we consider the problem of minimizing an indefinite quadratic function subject to a single indefinite quadratic constraint. A key difficulty with this problem is its nonconvexity. Using Lagrange duality, we show that under a mild assumption, this problem can be solved by solving a linearly constrained convex univariate minimization problem. Finally, the superior efficiency of the new approach compared to the known semidefinite relaxation and a known approach from the literature is demonstrated by solving several randomly generated test problems.     

Successive approximation technique for a class of large-scale NLP problems and its application to dynamic programming

In this paper, two successive approximation techniques are presented for a class of large-scale nonlinear programming problems with decomposable constraints and a class of high-dimensional discrete optimal control problems, respectively. It is shown that: (a) the accumulation point of the sequence produced by the first method is a Kuhn-Tucker point if the constraint functions are decomposable and if the uniqueness condition holds; (b) the sequence converges to an optimum solution if the objective function is strictly pseudoconvex and if the constraint functions are decomposable and quasiconcave; and (c) similar conclusions for the second method hold also for a class of discrete optimal control problems under some assumptions.     

On the global optimality of generalized trust region subproblems

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.