Algorithms for unconstrained global optimization of nonlinear (polynomial) programming problems: The single and multi-segment polynomial B-spline approach

We investigate the use of the polynomial B-spline form for unconstrained global optimization of multivariate polynomial nonlinear programming problems. We use the B-spline form for higher order approximation of multivariate polynomials. We first propose a basic algorithm for global optimization that uses several accelerating algorithms such as cut-off test and monotonicity test. We then propose an improved algorithm consisting of several additional ingredients, such as a new subdivision point selection rule and a modified subdivision direction selection rule. The performances of the proposed basic and improved algorithms are tested and compared on a set of 14 test problems under two test conditions. The results of the tests show the superiority of the improved algorithm with multi-segment B-spline over that of the single segment B-spline, in terms of the chosen performance metrics. We also compare the quality of the set of all global minimizers found using the proposed algorithms (basic & improved) with those using well-known solvers BARON and Gloptipoly, on a smaller set of four test problems. The problems in the latter set have multiple global minimizers. The results show the superiority of the proposed algorithms, in that they are able to capture all the global minimizers, whereas Gloptipoly and BARON fail to do so in some of the test problems.     

Fuzzy programming for mixed-integer optimization problems

Mixed-integer optimization problems belong to the group of NP-hard combinatorial problems. Therefore, they are difficult to search for global optimal solutions. Mixed-integer optimization problems are always described by precise mathematical programming models. However, many practical mixed-integer optimization problems have inherited a more or less imprecise nature. Under these circumstances, if we take into account the flexibility of the constraints and the fuzziness of the objectives, the original mixed-integer optimization problems can be formulated as fuzzy mixed-integer optimization problems. Mixed-integer hybrid differential evolution (MIHDE) is an evolutionary search algorithm which has been successfully applied to many complex mixed-integer optimization problems. In this article, a fuzzy mixed-integer mathematical programming model is developed to formulate the fuzzy mixed-integer optimization problem. In addition the MIHDE is introduced to solve the fuzzy mixed-integer programming problem. Finally, the illustrative example shows that satisfactory results can be obtained by the proposed method. This demonstrates that MIHDE can effectively handle fuzzy mixed-integer optimization problems.     

基于量子粒子群求解混合整数非线性规划

在经典微粒群算法的基础上提出一种有较高收敛性能的智能算法：量子粒子群（QPSO）算法。并用于求解混合整数非线性规划问题。实验室证明QPSO算法收敛性能好、速度快,为求解混合整数非线性规划开辟了新途径。     

Multiple-model based predictive control of nonlinear hybrid systems based on global optimization using the Bernstein polynomial approach

In this paper, we propose a Bernstein polynomial based global optimization algorithm for the optimal feedback control of nonlinear hybrid systems using a multiple-model approach. Specifically, we solve at every sampling instant a polynomial mixed-integer nonlinear programming problem arising in the model predictive control strategy. The proposed algorithm uses the Bernstein polynomial form in a branch-and-bound framework, with new ingredients such as branching for integer decision variables and fathoming for each subproblem in the branch-and-bound tree. The performance of the proposed algorithm is tested and compared with existing algorithms on a benchmark three-spherical tank system. The test results show the superior performance of the proposed algorithm.     

A mixed-integer programming approach for solving university course timetabling problems

Journal of Scheduling - This article presents a mixed-integer programming model for solving the university timetabling problem which considers the allocation of students to classes and the...     

A New Subdivision Algorithm for the Bernstein Polynomial Approach to Global Optimization

In this paper,an improved algorithm is proposed for unconstrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems.The proposed algorithm is based on the Bernstein polynomial approach.Novel features of the proposed algorithm are that it uses a new rule for the selection of the subdivision point,modified rules for the selection of the subdivision direction,and a new acceleration device to avoid some unnecessary subdivisions.The performance of the proposed algorithm is numerically tested on a collection of 16 test problems.The results of the tests show the proposed algorithm to be superior to the existing Bernstein algorithm in terms of the chosen performance metrics.     

Solving nonlinear polynomial systems in the barycentric Bernstein basis

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.     

A note on truss topology optimization under self-weight load: mixed-integer second-order cone programming approach

This study compares the performance of popular sampling methods for computer experiments using various performance measures to compare them. It is well known that the sample points, in the design space located by a sampling method, determine the quality of the meta-model generated based on expensive computer experiment (or simulation) results obtained at sample (or training) points. Thus, it is very important to locate the sample points using a sampling method suitable for the system of interest to be approximated. However, there is still no clear guideline for selecting an appropriate sampling method for computer experiments. As such, a sampling method, the optimal Latin hypercube design (OLHD), has been popularly used, and quasi-random sequences and the centroidal Voronoi tessellation (CVT) have begun to be noticed recently. Some literature on the CVT asserted that the performance of the CVT was better than that of the LHD, but this assertion seems unfair because those studies only employed space-filling performance measures in favor of the CVT. In this research, we performed the comparison study among the popular sampling methods for computer experiments (CVT, OLHD, and three quasi-random sequences) with employing both space-filling properties and a projective property as performance measures to fairly compare them. We also compared the root mean square error (RMSE) values of Kriging meta-models generated using the five sampling methods to evaluate their prediction performance. From the comparison results, we provided a guideline for selecting appropriate sampling methods for some systems of interest to be approximated.     

Maximizing the number of conflict-free aircraft using mixed-integer nonlinear programming

We address the conflict detection and resolution problem in air traffic control, where an aircraft conflict is a loss of separation between aircraft trajectories. Conflict avoidance is crucial to ensure flight safety and remains a challenging traffic control problem. We focus on speed control to separate aircraft and consider two approaches: (i) maximize the number of conflicts resolved and (ii) identify the largest set of conflict-free aircraft. Both problems are modeled using mixed-integer nonlinear programming and a tailored greedy algorithm is proposed for the latter. Computational efficiency is improved through a pre-processing algorithm which attempts to reduce the size of the conflict resolution models by detecting the existence of pairwise potential conflicts. Numerical results are provided after implementing the proposed models and algorithms on benchmark conflict resolution instances. The results highlight the benefits of using the proposed pre-processing step as well as the versatility and the efficiency of the proposed models.     

SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).     

求解非线性规划问题的混合粒子群算法

用粒子群算法求解非线性规划问题时不可避免的会产生不可行点,处理好不可行点是粒子群算法取得良好优化结果的关键。依据粒子的目标函数值与违反约束的程度提出了一种处理不可行点的合理选择方案,并运用融合差分演化的混合粒子群算法求解约束优化问题,数值实验表明该算法的有效性。     

A method of network programming in problems of nonlinear optimization

A method of network programming for solving problems of nonlinear optimization is used. A notion of dual problem is introduced. It is proved that a dual problem is a problem of convex programming. Necessary and sufficient conditions for optimality of dual problem of integer linear programming are obtained.     

A mixed-integer linear programming approach to the optimization of event-bus schedules: a scheduling application in the tourism sector

This paper deals with “The Enchanted Journey,” which is a daily event tour booked by Bollywood-film fans. During the tour, the participants visit original sites of famous Bollywood films at various locations in Switzerland; moreover, the tour includes stops for lunch and shopping. Each day, up to five buses operate the tour. For operational reasons, however, two or more buses cannot stay at the same location simultaneously. Further operative constraints include time windows for all activities and precedence constraints between some activities. The planning problem is how to compute a feasible schedule for each bus. We implement a two-step hierarchical approach. In the first step, we minimize the total waiting time; in the second step, we minimize the total travel time of all buses. We present a basic formulation of this problem as a mixed-integer linear program. We enhance this basic formulation by symmetry-breaking constraints, which reduces the search space without loss of generality. We report on computational results obtained with the Gurobi Solver. Our numerical results show that all relevant problem instances can be solved using the basic formulation within reasonable CPU time, and that the symmetry-breaking constraints reduce that CPU time considerably.     

线性约束非线性函数全局优化算法的研究

提出了一种适于处理线性约束条件下非线性规划问题的λ编码稳态遗传算法(λSSGA).首先对线性可行域进行凸分析后将原优化问题I转化为一个仅包含可行域极点信息的等价问题II.问题II具有小边界的约束条件,通过采用特定的凸交叉算子、交换变异算子和倒位算子可以保证算法在遗传操作的过程中不会产生无效的编码,而且能在概率意义上保证λ编码模式在整个可行解空间上充分可达.其次从理论上推导出了得到线性可行区域全部极点的方法,证明了问题I和问题II的等价性.仿真结果表明λSSGA算法在具有较快的收敛速度和精度的同时,还可以有效地维持群体的多样性,得到问题全局的最优解.     

An approach to the solution of nonlinear unconstrained optimization problems

An approach to reducing a constrained convex programming problem to an unconstrained optimization problem is considered. An initial internal feasible point is supposed to be specified. An equivalent unconstrained optimization problem is formulated in such a way that the calculated values of gradients (subgradients) of original functions do not violate the initial constraints. Properties of introduced functions are investigated. Convexity conditions are formulated for the unconstrained optimization problem. The results may by useful for the development of algorithms for solving constrained optimization problems.     

一类非线性两层规划问题的递阶优化解法

提出一种求解一类非线性两层规划问题的新方法．通过引入解耦向量将非线性两层规划问题分解为独立且易于求解的子问题，利用两级递阶结构第1级求解若干优化的子问题，而在第2级利用第1级求解的结果调整解耦向量．所提出的方法借助于分解一协调原理并按迭代方式最终求得问题的最优解．对于含整数的规划问题，通过连续化处理后也可按该方法方便地求解．算例表明所提出的算法是简便而有效的．     

GPU-based parallel solver via the Kantorovich theorem for the nonlinear Bernstein polynomial systems

This paper proposes a parallel solver for the nonlinear systems in Bernstein form based on subdivision and the Newton-Raphson method, where the Kantorovich theorem is employed to identify the existence of a unique root and guarantee the convergence of the Newton-Raphson iterations. Since the Kantorovich theorem accommodates a singular Jacobian at the root, the proposed algorithm performs well in a multiple root case. Moreover, the solver is designed and implemented in parallel on Graphics Processing Unit(GPU) with SIMD architecture; thus, efficiency for solving a large number of systems is improved greatly, an observation validated by our experimental results.     

On the solution of mixed-integer nonlinear programming models for computer aided molecular design

This paper addresses the efficient solution of computer aided molecular design (CAMD) problems, which have been posed as mixed-integer nonlinear programming models. The models of interest are those in which the number of linear constraints far exceeds the number of nonlinear constraints, and with most variables participating in the nonconvex terms. As a result global optimization methods are needed. A branch-and-bound algorithm (BB) is proposed that is specifically tailored to solving such problems. In a conventional BB algorithm, branching is performed on all the search variables that appear in the nonlinear terms. This translates to a large number of node traversals. To overcome this problem, we have proposed a new strategy for branching on a set of linear branchingfunctions, which depend linearly on the search variables. This leads to a significant reduction in the dimensionality of the search space. The construction of linear underestimators for a class of functions is also presented. The CAMD problem that is considered is the design of optimal solvents to be used as cleaning agents in lithographic printing.