Global optimization for the generalized polynomial sum of ratios problem

In this paper, a new deterministic global optimization algorithm is proposed for solving a fractional programming problem whose objective and constraint functions are all defined as the sum of generalized polynomial ratios, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. The proposed algorithm is based on reformulating the problem as a monotonic optimization problem, and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and numerical examples are given to illustrate the feasibility and efficiency of the present algorithm.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.     

Solving sum of quadratic ratios fractional programs via monotonic function

In this paper, a branch-reduce-bound algorithm is proposed for globally solving a sum of quadratic ratios fractional programming with nonconvex quadratic constraints. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. The proposed algorithm is based on reformulating the problem as a monotonic optimization problem, and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and the numerical experiments are given to show the feasibility of the proposed algorithm.     

Two-Machine Flow Shop Scheduling with Nonregular Criteria

We consider a two-machine flow shop problem with a common due date where the objective is to minimize the sum of functions which penalize early as well as tardy completion of jobs. Since the problem is NP-hard in the strong sense, we investigate some general properties of optimal schedules for the problem, we develop lower and upper bounds, derive dominance criteria, and propose an enumerative algorithm for finding an optimal schedule. The performance of the proposed algorithm together with the influence of the individual components is thoroughly discussed.     

Hybrid co-evolutionary particle swarm optimization and noising metaheuristics for the delay constrained least cost path problem

This paper presents a co-evolutionary particle swarm optimization (PSO) algorithm, hybridized with noising metaheuristics, for solving the delay constrained least cost (DCLC) path problem, i.e., shortest-path problem with a delay constraint on the total “cost” of the optimal path. The proposed algorithm uses the principle of Lagrange relaxation based aggregated cost. It essentially consists of two concurrent PSOs for solving the resulting minimization-maximization problem. The main PSO is designed as a hybrid PSO-noising metaheuristics algorithm for efficient global search to solve the minimization part of the DCLC-Lagrangian relaxation by finding multiple shortest paths between a source-destination pair. The auxiliary/second PSO is a co-evolutionary PSO to obtain the optimal Lagrangian multiplier for solving the maximization part of the Lagrangian relaxation problem. For the main PSO, a novel heuristics-based path encoding/decoding scheme has been devised for representation of network paths as particles. The simulation results on several networks with random topologies illustrate the efficiency of the proposed hybrid algorithm for the constrained shortest path computation problems.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

A labeling algorithm for the sensitivity ranges of the assignment problem

In view of the simplex-type algorithm, the assignment problem is inherently highly degenerate. It may be the optimal basis has changed, but the optimal assignment is unchanged when parameter variation occurs. Degeneracy then makes sensitivity analysis difficult, as well as makes the classical Type I range, which identifies the range the optimal basis unchanged, impractical. In this paper, a labeling algorithm is proposed to identify two other sensitivity ranges – Type II range and Type III range. The algorithm uses the reduced cost matrix, provided in the final results of most solution algorithms for AP, to determine the Type II range which reflects the stability of the current optimal assignment. Thus, the algorithm generates a streamlined situation from searching the optimal solution until performing the sensitivity analysis of the assignment problem. The Type III range, reflecting the flexibility of optimal decision making, can be obtained immediately after the Type II range is determined. Numerical examples are presented to demonstrate the algorithm.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

一种基于区间数多指标信息的FCM聚类算法

针对一类具有不确定性区间数多指标信息的聚类分析问题，依据传统的基于数值信息的FCM聚类算法的思路，提出了一种新的聚类分析算法。章首先描述了具有区间数多指标信息的聚类分析问题；其次给出了基于区间数多指标信息的关于最优划分和最优聚类中心确定的两个定理；然后给出了基于区间数多指标信息的FCM聚类算法的计算步骤。该算法的特点是聚类中心的表现形式为精确的数值，给出的两个定理说明了该聚类算法的收敛性。最后，通过给出一个算例说明了本给出的聚类算法。     

PSBH中的组合优化问题及其计算方法

本文介绍了具有部分位置信息的SBH杂交测序（Positional Sequencing by Hy-bridization,简称PSBH）实验所产生的一个重构DNA片断的组合优化问题,并讨论了该问题最优重构的计算问题.通过对PSBH提供的谱集及其位置信息的分析处理,我们获得了若干判定最优重构片断头尾的分支定界准则以及确定其非头尾位置最可能出现k-tuple的动态规划计算方法,并由此给出了该PSBH问题的一个新重构算法.该算法允许PSBH谱集含有一般杂交实验中常常可能出现探针错配所产生的正错误,并且仅仅假设PSBH的谱集、位置信息和位置长度是已知的,所以我们的算法具有更一般的适应性和实用性.此外,由于我们给出的算法能够极大地利用PSBH的谱集和位置信息所蕴含的信息确定最优重构片断头尾及其中间位置最可能出现的k-tuple,极大地减少了PSBH重构中的随意性,所以我们的算法也是有效的,模拟PSBH实验的计算结果验证了这一点.     

求线性比试和问题的确定性全局优化算法

为求线性比试和问题的全局最优解,本文给出了一个分支定界算法.通过一个等价问题和一个新的线性化松弛技巧,初始的非凸规划问题归结为一系列线性规划问题的求解.借助于这一系列线性规划问题的解,算法可收敛于初始非凸规划问题的最优解.算法的计算量主要是一些线性规划问题的求解.数值算例表明算法是切实可行的.     

A Lagrangian search method for the <Emphasis Type="Italic">P</Emphasis>-median problem

In this paper, we propose a novel algorithm for solving the classical P-median problem. The essential aim is to identify the optimal extended Lagrangian multipliers corresponding to the optimal solution of the underlying problem. For this, we first explore the structure of the data matrix in P-median problem to recast it as another equivalent global optimization problem over the space of the extended Lagrangian multipliers. Then we present a stochastic search algorithm to find the extended Lagrangian multipliers corresponding to the optimal solution of the original P-median problem. Numerical experiments illustrate that the proposed algorithm can effectively find a global optimal or very good suboptimal solution to the underlying P-median problem, especially for the computationally challenging subclass of P-median problems with a large gap between the optimal solution of the original problem and that of its Lagrangian relaxation.     

Optimal policy and stability regions for the single product periodic review inventory problem,with stationary demands

In this paper we present rules concerning the optimal policy and stability regions for the single product periodic review inventory problem with stationary demands, over a finite horizon. The key parameter to the whole study is the Lot-Sizing Index (LSI) introduced by Blackburn and Millen. Two algorithms are presented. The first one constructs stability regions which are expressed as intervals of the LSI parameter, covering the whole range of its values. The proposed algorithm is very simple to understand and implement, and most importantly, it provides a solution table which can be used by the decision maker to easily determine the optimal policy for any problem with a given horizon and any possible combination of its cost parameters, namely any LSI value. The second proposed algorithm determines the optimal policy for any given LSI value; it constitutes a completely different approach to that of the Wagner–Whitin algorithm and requires very little computational effort.     

A new approach to the weighted peak-constrained least-square error FIR digital filter optimal design problem

This paper deals with the design of linear-phase finite impulse response (FIR) digital filters using weighted peak-constrained least-squares (PCLS) optimization. The PCLS error design problem is formulated as a quadratically constrained quadratic semi-infinite programming problem. An exchange algorithm with a new exchange rule is proposed to solve the problem. The algorithm provides the approximate optimal solution after a finite number of iterations. In particular, the subproblem solved at each iteration is a quadratically constrained quadratic programming. We can rewrite it as a conic optimization problem solvable in polynomial time. For illustration, numerical examples are solved using the proposed algorithm.     

An algorithm for local continuous optimization of traffic signals

In this paper, an algorithm for sensitivity analysis for equilibrium traffic network flows with link interferences is proposed. Based on this sensitivity analysis algorithm, a general algorithm is provided for solving the optimal design and management problems for traffic networks. In particular, this algorithm is applied to the optimal traffic signal setting problem. A numerical example is given to demonstrate the effectiveness of our algorithm.     

Mean–variance optimal trading problem subject to stochastic dominance constraints with second order autoregressive price dynamics

The efficient modeling of execution price path of an asset to be traded is an important aspect of the optimal trading problem. In this paper an execution price path based on the second order autoregressive process is proposed. The proposed price path is a generalization of the existing first order autoregressive price path in literature. Using dynamic programming method the analytical closed form solution of unconstrained optimal trading problem under the second order autoregressive process is derived. However in order to incorporate non-negativity constraints in the problem formulation, the optimal static trading problems under second order autoregressive price process are formulated. For a risk neutral investor, the optimal static trading problem of minimizing expected execution cost subject to non-negativity constraints is formulated as a quadratic programming problem. Whereas, for a risk averse investor the variance of execution cost is considered as a measure for the timing risk, and the mean–variance problem is formulated. Moreover, the optimal static trading problem subject to stochastic dominance constraints with mean–variance static trading strategy as the reference strategy is studied. Using Static approximation method the algorithm to solve proposed optimal static trading problems is presented. With numerical illustrations conducted on simulated data and the real market data, the significance of second order autoregressive price path, and the optimal static trading problems is presented.     

An algorithm for optimizing over the weakly-efficient set

In this paper a bisecting search algorithm is developed for solving the problem (P) of optimizing a linear function over the set of weakly-efficient solutions of a multiple objective linear program. We show that problem (P) can arise in a variety of practical situations. The algorithm for solving problem (P) is guaranteed to find an optimal or approximately-optimal solution for the problem in a finite number of steps. Using a Fortran computer code called Conmin as an aid, we have solved ten test problems using our proposed algorithm. This preliminary computational experience seems to indicate that the algorithm is quite practical for relatively small problems.     

Single Machine Scheduling and Selection to Minimize Total Flow Time with Minimum Number Tardy

Emmons considered the problem of sequencing N jobs on a single machine to minimize total flow time with the minimum number of tardy jobs. He proposed an effective branch-and-bound algorithm for this problem. In this paper, we show that Emmons' algorithm can be extended to a more difficult scheduling problem which includes an optimal selection of jobs as well.     

A smoothing SQP method for nonlinear programs with?stability constraints arising from power systems

This paper investigates a new class of optimization problems arising from power systems, known as nonlinear programs with stability constraints (NPSC), which is an extension of ordinary nonlinear programs. Since the stability constraint is described generally by eigenvalues or norm of Jacobian matrices of systems, this results in the semismooth property of NPSC problems. The optimal conditions of both NPSC and its smoothing problem are studied. A smoothing SQP algorithm is proposed for solving such optimization problem. The global convergence of algorithm is established. A numerical example from optimal power flow (OPF) is done. The computational results show efficiency of the new model and algorithm.