Extensions of Dinkelbach's algorithm for solving non-linear fractional programming problems

Dinkelbach's algorithm was developed to solve convex fractinal programming. This method achieves the optimal solution of the optimisation problem by means of solving a sequence of non-linear convex programming subproblems defined by a parameter. In this paper it is shown that Dinkelbach's algorithm can be used to solve general fractional programming. The applicability of the algorithm will depend on the possibility of solving the subproblems. Dinkelbach's extended algorithm is a framework to describe several algorithms which have been proposed to solve linear fractional programming, integer linear fractional programming, convex fractional programming and to generate new algorithms. The applicability of new cases as nondifferentiable fractional programming and quadratic fractional programming has been studied. We have proposed two modifications to improve the speed-up of Dinkelbachs algorithm. One is to use interpolation formulae to update the parameter which defined the subproblem and another truncates the solution of the suproblem. We give sufficient conditions for the convergence of these modifications. Computational experiments in linear fractional programming, integer linear fractional programming and non-linear fractional programming to evaluate the efficiency of these methods have been carried out.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

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.     

A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization

We present numerical results of a comparative study of codes for nonlinear and nonconvex mixed-integer optimization. The underlying algorithms are based on sequential quadratic programming (SQP) with stabilization by trust-regions, linear outer approximations, and branch-and-bound techniques. The mixed-integer quadratic programming subproblems are solved by a branch-and-cut algorithm. Second order information is updated by a quasi-Newton update formula (BFGS) applied to the Lagrange function for continuous, but also for integer variables. We do not require that the model functions can be evaluated at fractional values of the integer variables. Thus, partial derivatives with respect to integer variables are replaced by descent directions obtained from function values at neighboring grid points, and the number of simulations or function evaluations, respectively, is our main performance criterion to measure the efficiency of a code. Numerical results are presented for a set of 100 academic mixed-integer test problems. Since not all of our test examples are convex, we reach the best-known solutions in about 90 % of the test runs, but at least feasible solutions in the other cases. The average number of function evaluations of the new mixed-integer SQP code is between 240 and 500 including those needed for one- or two-sided approximations of partial derivatives or descent directions, respectively. In addition, we present numerical results for a set of 55 test problems with some practical background in petroleum engineering.     

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。     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.     

Outer-approximation algorithms for nonsmooth convex MINLP problems

In this work, we combine outer-approximation (OA) and bundle method algorithms for dealing with mixed-integer non-linear programming (MINLP) problems with nonsmooth convex objective and constraint functions. As the convergence analysis of OA methods relies strongly on the differentiability of the involved functions, OA algorithms may fail to solve general nonsmooth convex MINLP problems. In order to obtain OA algorithms that are convergent regardless the structure of the convex functions, we solve the underlying OA’s non-linear subproblems by a specialized bundle method that provides necessary information to cut off previously visited (non-optimal) integer points. This property is crucial for proving (finite) convergence of OA algorithms. We illustrate the numerical performance of the given proposal on a class of hybrid robust and chance-constrained problems that involve a random variable with finite support.     

An algorithm of sequential systems of linear equations for nonlinear optimization problems with arbitrary initial point

For current sequential quadratic programming (SQP) type algorithms, there exist two problems: (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using ε-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented. This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above. Project partly supported by the National Natural Science Foundation of China and Tianyuan Foundation of China.     

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

Lift-and-project cuts for convex mixed integer nonlinear programs

We describe a computationally effective method for generating lift-and-project cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can be obtained from solving a cut-generating nonlinear program. Using this procedure, we are able to approximately optimize over the rank one lift-and-project closure for a variety of convex MINLP instances. The results indicate that lift-and-project cuts have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs significantly reduces the total solution time for many instances. We also demonstrate that combining lift-and-project cuts with an extended formulation that exploits separability of convex functions yields significant improvements in both relaxation bounds and the time to calculate the relaxation. Overall, these results suggest that with an effective separation routine, like the one proposed here, lift-and-project cuts may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.     

凸规划的新算法

对一般凸目标函数和一般凸集约束的凸规划问题新解法进行探讨,它是线性规划一种新算法的扩展和改进,此算法的基本思想是在规划问题的可行域中由所建－的一个切割面到另一个切割面的不断推进来求取最优的。文章对目标函数是二次的且约束是一般凸集和二次目标函数且约束是线性的情形,给出了更简单的算法。     

Unbounded convex sets for non-convex mixed-integer quadratic programming

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied convex sets. Several classes of valid and facet-inducing inequalities are also derived.     

Models and solution techniques for production planning problems with increasing byproducts

We consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete-time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MILP) approximation and relaxation models of this nonconvex MINLP, and we derive modifications that strengthen the linear programming relaxations of these models. We also introduce nonlinear programming formulations to choose piecewise-linear approximations and relaxations of multiple functions that share the same domain and use the same set of break points in the domain. We conclude with computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MILP approximation and relaxation models can be used to obtain provably near-optimal solutions for large instances of this nonconvex MINLP. Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.     

An Improved Bernstein Global Optimization Algorithm for MINLP Problems with Application in Process Industry

We present an improved Bernstein global optimization algorithm to solve polynomial mixed-integer nonlinear programming (MINLP) problems. The algorithm is of branch-and-bound type, and uses the Bernstein form of the polynomials for the global optimization. The new ingredients in the algorithm include a modified subdivision procedure, a vectorized Bernstein cut-off test and a new branching rule for the decision variables. The performance of the improved algorithm is tested and compared with earlier reported Bernstein global optimization algorithm (to solve polynomial MINLPs) and with several state-of-the-art MINLP solvers on a set of 19 test problems. The results of the tests show the superiority of the improved algorithm over the earlier reported Bernstein algorithm and the state-of-the-art solvers in terms of the chosen performance metrics. Similarly, efficacy of the improved algorithm in handling a real-world MINLP problem is brought out via a trim-loss minimization problem from the process industry.     

Iterative linear programming solution of convex programs

An iterative linear programming algorithm for the solution of the convex programming problem is proposed. The algorithm partially solves a sequence of linear programming subproblems whose solution is shown to converge quadratically, superlinearly, or linearly to the solution of the convex program, depending on the accuracy to which the subproblems are solved. The given algorithm is related to inexact Newton methods for the nonlinear complementarity problem. Preliminary results for an implementation of the algorithm are given.This material is based on research supported by the National Science Foundation, Grants DCR-8521228 and CCR-8723091, and by the Air Force Office of Scientific Research, Grant AFOSR-86-0172. The author would like to thank Professor O. L. Mangasarian for stimulating discussions during the preparation of this paper.     

Hyperplane method for reachable state estimation for linear time-invariant systems

A numerical algorithm is presented for generating inner and outer approximations for the set of reachable states for linear time-invariant systems. The algorithm is based on analytical results characterizing the solutions to a class of optimization problems which determine supporting hyperplanes for the reachable set. Explicit bounds on the truncation error for the finite-time case yield a set of so-called -supporting hyperplanes which can be generated to approximate the infinite-time reachable set within an arbitrary degree of accuracy. At the same time, an inner approximation is generated as the convex hull of points on the boundary of the finite-time reachable set. Numerical results are presented to illustrate the hyperplane method. The concluding section discusses directions for future work and applications of the method to problems in trajectory planning in servo systems.This research was supported in part by Digital Equipment Corporation through the American Electronics Association Fellowship Loan Program and by the National Science Foundation under Grant No. ECS-84-04607.