Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming

In this paper, we present a new hybrid algorithm for convex Mixed Integer Nonlinear Programming (MINLP). The proposed hybrid algorithm is an improved version of the classical nonlinear branch-and-bound (BB) procedure, where the enhancements are obtained with the application of the outer approximation algorithm on some nodes of the enumeration tree. The two methods are combined in such a way that each one collaborates to the convergence of the other. Computational experiments with benchmark instances of the MINLP problem show the good performance of the proposed algorithm, which is compared to the outer approximation algorithm, the nonlinear BB algorithm and the hybrid algorithm implemented in the solver Bonmin.     

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.     

Logarithmic barrier decomposition-based interior point methods for stochastic symmetric programming

We introduce and study two-stage stochastic symmetric programs with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a Benders’ decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao’s results [G. Zhao, A log barrier method with Benders’ decomposition for solving two-stage stochastic linear programs, Math. Program. Ser. A 90 (2001) 507–536] for two-stage stochastic linear programs, and Mehrotra and Özevin’s results [S. Mehrotra, M.G. Özevin, Decomposition-based interior point methods for two-stage stochastic semidefinite programming, SIAM J. Optim. 18 (1) (2007) 206–222] for two-stage stochastic semidefinite programs.     

Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems.In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver.A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of up to 3 improvement over branch-and-bound is observed.     

A numerical study of MIDACO on 100 MINLP benchmarks

This article presents a numerical study on MIDACO, a new global optimization software for mixed integer non-linear programming (MINLP) based on ant colony optimization and the oracle penalty method. Extensive and rigorous numerical tests on a set of 100 non-convex MINLP benchmark problems from the open literature are performed. Results obtained by MIDACO are directly compared to results by a recent study of state-of-the-art deterministic MINLP software on the same test set. Further comparisons with an established MINLP software is undertaken in addition. This study shows that MIDACO is not only competitive to the established MINLP software, but can even outperform those in terms of the number of global optimal solutions found. Moreover, the parallelization capabilities of MIDACO enable it to be even competitive to deterministic software regarding the amount of (serial processed) function evaluation, while the black-box capabilities of MIDACO offer an intriguing new robustness for MINLP.     

Variable neighborhood search: Principles and applications

Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a two-level VNS, called variable neighborhood decomposition search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear to have been applied before.     

解大型无约束优化的新的非单调简单模型BB-TR方法（英文）

The trust region(TR) method for optimization is a class of effective methods.The conic model can be regarded as a generalized quadratic model and it possesses the good convergence properties of the quadratic model near the minimizer.The Barzilai and Borwein(BB) gradient method is also an effective method,it can be used for solving large scale optimization problems to avoid the expensive computation and storage of matrices.In addition,the BB stepsize is easy to determine without large computational efforts.In this paper,based on the conic trust region framework,we employ the generalized BB stepsize,and propose a new nonmonotone adaptive trust region method based on simple conic model for large scale unconstrained optimization.Unlike traditional conic model,the Hessian approximation is an scalar matrix based on the generalized BB stepsize,which resulting a simple conic model.By adding the nonmonotone technique and adaptive technique to the simple conic model,the new method needs less storage location and converges faster.The global convergence of the algorithm is established under certain conditions.Numerical results indicate that the new method is effective and attractive for large scale unconstrained optimization problems.     

Siting and Sizing of Facilities under Probabilistic Demands

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A Mixed-Integer Nonlinear Programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.     

Two linear approximation algorithms for convex mixed integer nonlinear programming

We present two new algorithms for convex Mixed Integer Nonlinear Programming (MINLP), both based on the well known Extended Cutting Plane (ECP) algorithm proposed by Weterlund and Petersson. Our first algorithm, Refined Extended Cutting Plane (RECP), incorporates additional cuts to the MILP relaxation of the original problem, obtained by solving linear relaxations of NLP problems considered in the Outer Approximation algorithm. Our second algorithm, Linear Programming based Branch-and-Bound (LP-BB), applies the strategy of generating cuts that is used in RECP, to the linear approximation scheme used by the LP/NLP based Branch-and-Bound algorithm. Our computational results show that RECP and LP-BB are highly competitive with the most popular MINLP algorithms from the literature, while keeping the nice and desirable characteristic of ECP, of being a first-order method.

     

A global MINLP approach to symbolic regression

Symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values. As a result, the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition, subtraction, multiplication, and division, among others. Currently, state-of-the-art symbolic regression methods leverage genetic algorithms and adaptive programming techniques. Genetic algorithms lack optimality certifications and are typically stochastic in nature. In contrast, we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem. We present a mixed-integer nonlinear programming (MINLP) formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries. We demonstrate this symbolic regression technique using an array of experiments based upon literature instances. We then use a set of 24 MINLPs from symbolic regression to compare the performance of five local and five global MINLP solvers. Finally, we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature.     

Barzilai–Borwein method with variable sample size for stochastic linear complementarity problems

A smoothing method for solving stochastic linear complementarity problems is proposed. The expected residual minimization reformulation of the problem is considered, and it is approximated by the sample average approximation (SAA). The proposed method is based on sequential solving of a sequence of smoothing problems where each of the smoothing problems is defined with its own sample average approximation. A nonmonotone line search with a variant of the Barzilai–Borwein (BB) gradient direction is used for solving each of the smoothing problems. The BB search direction is efficient and low cost, particularly suitable for nonmonotone line search procedure. The variable sample size scheme allows the sample size to vary across the iterations and the method tends to use smaller sample size far away from the solution. The key point of this strategy is a good balance between the variable sample size strategy, the smoothing sequence and nonmonotonicity. Eventually, the maximal sample size is used and the SAA problem is solved. Presented numerical results indicate that the proposed strategy reduces the overall computational cost.     

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.     

Siting and Sizing of Facilities under Probabilistic Demands

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A Mixed-Integer Nonlinear Programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.     

An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré.     

A parallel algorithm for two-staged two-dimensional fixed-orientation cutting problems

In this paper, we study the two-staged two-dimensional fixed-orientation cutting problem. We investigate the use of the parallel beam search algorithm for approximately solving the problem. The beam-search can be viewed as a truncated tree-search in which a subset of generated nodes are investigated. The proposed approach tries to explore some of these nodes in parallel by applying a master-slave paradigm. The master processor serves to guide the search-resolution by using a best-first search strategy for selecting the successive sets of nodes, called elite nodes. Whereas each slave processor develops the search tree and updates the global list of the master processor in an asynchronous manner. Each processor is based on combining a partial lower bound and a complementary upper bound, obtained by solving a series of bounded knapsack problems. The proposed method is analyzed computationally on a set of benchmark instances of the literature and their results are compared to those provided by existing algorithms. Encouraging and new results have been obtained.     

An efficient strategy for the activation of MIP relaxations in a multicore global MINLP solver

Solving mixed-integer nonlinear programming (MINLP) problems to optimality is a NP-hard problem, for which many deterministic global optimization algorithms and solvers have been recently developed. MINLPs can be relaxed in various ways, including via mixed-integer linear programming (MIP), nonlinear programming, and linear programming. There is a tradeoff between the quality of the bounds and CPU time requirements of these relaxations. Unfortunately, these tradeoffs are problem-dependent and cannot be predicted beforehand. This paper proposes a new dynamic strategy for activating and deactivating MIP relaxations in various stages of a branch-and-bound algorithm. The primary contribution of the proposed strategy is that it does not use meta-parameters, thus avoiding parameter tuning. Additionally, this paper proposes a strategy that capitalizes on the availability of parallel MIP solver technology to exploit multicore computing hardware while solving MINLPs. Computational tests for various benchmark libraries reveal that our MIP activation strategy works efficiently in single-core and multicore environments.     

A novel three-phase trajectory informed search methodology for global optimization

A new deterministic method for solving a global optimization problem is proposed. The proposed method consists of three phases. The first phase is a typical local search to compute a local minimum. The second phase employs a discrete sup-local search to locate a so-called sup-local minimum taking the lowest objective value among the neighboring local minima. The third phase is an attractor-based global search to locate a new point of next descent with a lower objective value. The simulation results through well-known global optimization problems are shown to demonstrate the efficiency of the proposed method.     

基于自适应遗传算法的逐次超松驰迭代法

确定逐次超松驰迭代法中的最佳松驰因子,迄今,人们还没有给出一可行实用的方法.利用自适应遗传算法全局搜索性能、并行性及其遗传操作,构造出近似确定最佳松驰因子的一种自适应进化方法,并由此得到一近似确定ω功能的自适应逐次超松驰迭代算法.数值算例表明,该算法在求解线性方程组中是可行的,实用和快捷的.     

Parallel Information Algorithm with Local Tuning for Solving Multidimensional GO Problems

In this paper we propose a new parallel algorithm for solving global optimization (GO) multidimensional problems. The method unifies two powerful approaches for accelerating the search: parallel computations and local tuning on the behavior of the objective function. We establish convergence conditions for the algorithm and theoretically show that the usage of local information during the global search permits to accelerate solving the problem significantly. Results of numerical experiments executed with 100 test functions are also reported.     

A new adaptive Barzilai and Borwein method for unconstrained optimization

In this paper we view the Barzilai and Borwein (BB) method from a new angle, and present a new adaptive Barzilai and Borwein (NABB) method with a nonmonotone line search for general unconstrained optimization. In the proposed method, the scalar approximation to the Hessian matrix is updated by the Broyden class formula to generate an adaptive stepsize. It is remarkable that the new stepsize is chosen adaptively in the interval which contains the two well-known BB stepsizes. Moreover, for the negative curvature direction, a strategy for the choice of the stepsize is designed to accelerate the convergence rate of the NABB method. Furthermore, we apply the NABB method without any line search to strictly convex quadratic minimization. The numerical experiments show the NABB method is very promising.