Optimal design of multi-response experiments using semi-definite programming

Optimal design of multi-response experiments for estimating the parameters of multi-response linear models is a challenging problem. The main drawback of the existing algorithms is that they require the solution of many optimization problems in the process of generating an optimal design that involve cumbersome manual operations. Furthermore, all the existing methods generate approximate design and no method for multi-response n-exact design has been cited in the literature. This paper presents a unified formulation for multi-response optimal design problem using Semi-Definite Programming (SDP) that can generate D-, A- and E-optimal designs. The proposed method alleviates the difficulties associated with the existing methods. It solves a one-shot optimization model whose solution selects the optimal design points among all possible points in the design space. We generate both approximate and n-exact designs for multi-response models by solving SDP models with integer variables. Another advantage of the proposed method lies in the amount of computation time taken to generate an optimal design for multi-response models. Several test problems have been solved using an existing interior-point based SDP solver. Numerical results show the potentials and efficiency of the proposed formulation as compared with those of other existing methods. The robustness of the generated designs with respect to the variance-covariance matrix is also investigated.     

Semi-definite programming-based method for security-constrained unit commitment with operational and optimal power flow constraints

Considering the economics and securities for the operation of a power system, a semi-definite programming (SDP) model for the security-constrained unit commitment (SCUC) problem is described here, which is directly solved by the interior-point method for SDP within the polynomial times. The proposed method is promising for the SCUC problems because of its excellent convergence and the ability of handling the non-covex integer variables. No model decomposition and initial relaxation are needed when applying the SDP-based method. When the solution contains minor mismatches in the integer variables, a simple rounding strategy is used to correct the non-integer into integer efficiently. Different test cases from 6 to 118 buses over a 24 h horizon are presented. Extensive numerical simulations have shown that the proposed method is capable of obtaining the optimal UC schedules without any network and bus voltage violations, and minimising the operation cost as well.     

Exploiting group symmetry in truss topology optimization

We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in Ohsaki et al. (Comp. Meth. Appl. Mech. Eng. 180:203–217, 1999). We show how one may automatically obtain symmetric designs, by eliminating the ‘redundant’ symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.     

An improved sum-of-disjoint-products technique for the symbolic network reliability analysis with known minimal paths

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. The sum-of-disjoint products technique (SDP) is a major fundamental tool for evaluating stochastic network reliability. In this study, a new SDP based on some intuitive properties that characterize the structure of minimal paths (MPs), and the relationships between MPs and subpaths are developed to improved SDP. The proposed SDP is easier to understand and implement, and better than the existing best- known SDP based algorithms under some special situation. The correctness of the proposed algorithm will be analyzed and proven. One bench example is illustrated to show how the network reliability with known MPs is determined using the proposed SDP.     

The adjuvants aerosil 200 and Gelita-Sol-P influence on the technological characteristics of spray-dried powders from Passiflora edulis var. flavicarpa

Passiflora edulis (passionflower) is a plant widely used in the Brazilian popular medicine and phytopharmaceutical industry for its sedative activity. This work refers to the development of spray-dried powders (SDPs) from the 40% ethanolic extractive solution of P. edulis aerial parts. The SDPs were prepared with a Büchi 190 Mini-spray dryer using as drying adjuvants Aerosil 200 alone (SDP1), an Aerosil 200: Gelita-Sol-P (1:1) mixture (SDP2) and an Aerosil 200:Gelita-Sol-P (1:3) mixture (SDP3). All the powders were obtained using 40 parts adjuvant and 60 parts extract dried residue. The comparison criteria applied were particle size distribution, hygroscopicity at 95%, 60%, and 35% relative humidity (RH), as well as the flavonoid process recovery. The particle size distributions were analyzed by way of (a) normal distribution parameters, (b) the RRSB grid and (c) considering diameter values in terms of an equivalent sphere. All the powders presented nonnormal distribution, and the RRSB analysis appeared to be, therefore, the analysis method of choice. The total flavonoid recovery was around 80%, and it was practically not affected by the SDP1, SDP2, and SDP3 compositions. At the 60% and 90% RH atmospheres, the SDP3 and SDP2 moisture uptakes were higher than that of the SDP1. All the formulations were, on the contrary, stable at 35% RH, showing a slight moisture loss tendency. The results showed in sum that the SDP prepared using Aerosil 200 as the drying adjuvant alone presented the best technological characteristics of all.     

The Adjuvants Aerosil 200 and Gelita-Sol-P Influence on the Technological Characteristics of Spray-Dried Powders from Passiflora edulis var. flavicarpa

Passiflora edulis (passionflower) is a plant widely used in the Brazilian popular medicine and phytopharmaceutical industry for its sedative activity. This work refers to the development of spray-dried powders (SDPs) from the 40% ethanolic extractive solution of P. edulis aerial parts. The SDPs were prepared with a Büchi 190 Mini-spray dryer using as drying adjuvants Aerosil 200^® alone (SDP1), an Aerosil 200:Gelita-Sol-P^® (1:1) mixture (SDP2) and an Aerosil 200:Gelita-Sol-P (1:3) mixture (SDP3). All the powders were obtained using 40 parts adjuvant and 60 parts extract dried residue. The comparison criteria applied were particle size distribution, hygroscopicity at 95%, 60%, and 35% relative humidity (RH), as well as the flavonoid process recovery. The particle size distributions were analyzed by way of (a) normal distribution parameters, (b) the RRSB grid and (c) considering diameter values in terms of an equivalent sphere. All the powders presented nonnormal distribution, and the RRSB analysis appeared to be, therefore, the analysis method of choice. The total flavonoid recovery was around 80%, and it was practically not affected by the SDP1, SDP2, and SDP3 compositions. At the 60% and 90% RH atmospheres, the SDP3 and SDP2 moisture uptakes were higher than that of the SDP1. All the formulations were, on the contrary, stable at 35% RH, showing a slight moisture loss tendency. The results showed in sum that the SDP prepared using Aerosil 200 as the drying adjuvant alone presented the best technological characteristics of all.     

An efficient interior-point algorithm with new non-monotone line search filter method for nonlinear constrained programming

An efficient primal-dual interior-point algorithm using a new non-monotone line search filter method is presented for nonlinear constrained programming, which is widely applied in engineering optimization. The new non-monotone line search technique is introduced to lead to relaxed step acceptance conditions and improved convergence performance. It can also avoid the choice of the upper bound on the memory, which brings obvious disadvantages to traditional techniques. Under mild assumptions, the global convergence of the new non-monotone line search filter method is analysed, and fast local convergence is ensured by second order corrections. The proposed algorithm is applied to the classical alkylation process optimization problem and the results illustrate its effectiveness. Some comprehensive comparisons to existing methods are also presented.     

二阶锥规划的非精确不可行内点法

二阶锥规划在工程、控制、金融等领域具有广泛的应用.本文研究一种求解二阶锥规划的非精确不可行内点法.该算法的基本思想是首先定义不可行中心路径及其邻域,然后通过求解一个非线性方程组得到非精确的搜索方向,再取一个合适的步长,使得新的迭代点落在不可行中心路径的邻域内.该算法不要求初始点和迭代点位于严格可行解集内.在适当的假设条件下证明了算法只需迭代O(√n ln(1ε))次就可以找到问题的ε-近似解.     

电路二等分问题的强化半定规划松弛

将表示电路的超图转化成带权值的无向图，从而将电路二等分问题转化成图的划分问题，图的划分问题存在已知的半定规划松弛，在此半定规划松弛基础上增加两个非线性结束，得到了强化半定规划松弛，定理和数值试验保证了强化半定规划松弛给出原问题一个更好的下界。     

A Primal-Dual Interior-Point Method to Solve the Optimal Power Flow Dispatching Problem

This paper presents a primal-dual path-following interior-point method for the solution of the optimal power flow dispatching (OPFD) problem. The underlying idea of most path-following algorithms is relatively similar: starting from the Fiacco-McCormick barrier function, define the central path and loosely follow it to the optimum solution. Several primal-dual methods for OPF have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. Nevertheless, there are substantial variations in some crucial details which include the formulation of the non-linear problem, the associated linear system, the linear algebraic procedure to solve this system, the line search, strategies for adjusting the centring parameter, estimating higher order correction terms for the homotopy path, and the treatment of indefiniteness. This paper discusses some of the approaches that were undertaken in implementing a specific primal-dual method for OPFD. A comparison is carried out with previous research on interior-point methods for OPF. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.     

Application research on semi-definite programming optimized support vector machines

The kernel function optimization is the key issues to address when using the support vector machine (SVM) algorithm. To solve the parameter selection for the SVM, a semi-definite programming optimized SVM (SDP-SVM) algorithm is proposed in this paper. The steps of the algorithm are described, and the optimization of the kernel function is shown using an SDP method. The SDP method is used to find the best parameter of SVM. The heart_scale data in the University of California Irvine database are then simulated using the SDP-SVM model. The experimental results shows that the generalization capability and the classification accuracy of the SDP-SVM algorithm have been greatly improved. A variety of strip-steel surface defect images from actual production are classified using the SDP-SVM algorithm, and the results show that the classification method of the SDP-SVM algorithm has high classification accuracy, strong practicability, and a wide variety of application prospects.     

修正一类非单调线性互补问题的宽邻域路径跟踪算法

通过对非单调线性互补问题所提出的一种内点算法进行分析,指出了算法中存在的关键性错误,在此基础上给出了求解一类非单调线性互补问题的宽邻域路径跟踪算法,给出了作为复杂性分析基础的两个重要关系式的正确表达式,并克服了由此带来的在收敛性分析中的一系列困难,成功地将线性规划问题的宽邻域内点算法,推广到非单调线性互补问题,讨论了算法的迭代复杂性。特别对于单调线性互补问题,得到了目前宽邻域内点算法迭代复杂性的最好结果。     

A stochastic dynamic programming model for stream water quality management

This paper deals with development of a seasonal fraction-removal policy model for waste load allocation in streams addressing uncertainties due to randomness and fuzziness. A stochastic dynamic programming (SDP) model is developed to arrive at the steady-state seasonal fraction-removal policy. A fuzzy decision model (FDM) developed by us in an earlier study is used to compute the system performance measure required in the SDP model. The state of the system in a season is defined by streamflows at the headwaters during the season and the initial DO deficit at some pre-specified checkpoints. The random variation of streamflows is included in the SDP model through seasonal transitional probabilities. The decision vector consists of seasonal fraction-removal levels for the effluent dischargers. Uncertainty due to imprecision (fuzziness) associated with water quality goals is addressed using the concept of fuzzy decision. Responses of pollution control agencies to the resulting end-of-season DO deficit vector and that of dischargers to the fraction-removal levels are treated as fuzzy, and modelled with appropriate membership functions. Application of the model is illustrated with a case study of the Tungabhadra river in India.     

Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion

The incremental problem for quasistatic elastoplastic analysis with the von?Mises yield criterion is discussed within the framework of the second-order cone programming (SOCP). We show that the associated flow rule under the von?Mises yield criterion with the linear isotropic/kinematic hardening is equivalently rewritten as a second-order cone complementarity problem. The minimization problems of the potential energy and the complementary energy for incremental analysis are then formulated as the primal-dual pair of SOCP problems, which can be solved with a primal-dual interior-point method. To enhance numerical performance of tracing an equilibrium path, we propose a warm-start strategy for a primal-dual interior-point method based on the primal-dual penalty method. In this warm-start strategy, we solve a penalized SOCP problem to find the equilibrium solution at the current loading step. An advanced initial point for solving this penalized SOCP problem is defined by using information of the solution at the previous loading step.     

Global optimization for signomial discrete programming problems in engineering design

Signomial discrete programming (SDP) problems arise frequently in a variety of real applications. Although many optimization techniques have been developed to solve an SDP problem, they use too many binary variables to reformulate the problem for finding a globally optimal solution or can only derive a local or an approximate solution. This article proposes a global optimization method to solve an SDP problem by integrating an efficient linear expression of single variable discrete functions and convexification techniques. An SDP problem can be converted into a convex mixed-integer programming problem solvable to obtain a global optimum. Several illustrative examples are also presented to demonstrate the usefulness and effectiveness of the proposed method.     

Interior-Point Methods in l 1 Optimal Sparse Representation Algorithms for Harmonic Retrieval

Atomic decomposition is an alternative method for frequency detection in harmonic signals. This type of method produces very concentrated solutions with few nonzero components. It can be used as an alternative to traditional approaches, such as, principal-components frequency estimation methods. In this paper, we consider the basis pursuit principle to find the representation (frequency) coefficients of a harmonic signal by minimizing the l ₁ norm. For the l ₁ minimization, we compare two interior-point methods. A primal-dual method, which consists of the perturbed optimality conditions of the linear program, results in solutions that are more accurate and sparse than using a primal (affine scaling) method to solve the same linear program. We contrast the solutions obtained by the interior-point methods using the size of the given data and a bound for perfect recovery of the harmonic signals to establish the better performance of the primal-dual method. In addition, experimental results are shown.     

Shakedown analysis with multidimensional loading spaces

A numerical method for the computation of shakedown loads of structures subjected to varying thermal and mechanical loading is proposed for the case of multidimensional loading spaces. The shakedown loading factors are determined based on the lower bound direct method using the von Mises yield criterion. The resulting nonlinear convex optimization problem is solved by use of the interior-point method. Although the underlying theory allows for the consideration of arbitrary numbers of loadings in principle, until now applications have been restricted to special cases, where either one or two loads vary independently. In this article, former formulations of the optimization problem are generalized for the case of arbitrary numbers of loadings. The method is implemented into an interior-point algorithm specially designed for this method. For illustration, numerical results are presented for a three-dimensional loading space applied to a square plate with a central circular hole.     

Finding multiple optimal solutions of signomial discrete programming problems with free variables

With the increasing reliance on mathematical programming based approaches in various fields, signomial discrete programming (SDP) problems occur frequently in real applications. Since free variables are often introduced to model problems and alternative optima are practical for decision making among multiple strategies, this paper proposes a generalized method to find multiple optimal solutions of SDP problems with free variables. By means of variable substitution and convexification strategies, an SDP problem with free variables is first converted into another convex mixed-integer nonlinear programming problem solvable to obtain an exactly global optimum. Then a general cut is utilized to exclude the previous solution and an algorithm is developed to locate all alternative optimal solutions. Finally, several illustrative examples are presented to demonstrate the effectiveness of the proposed approach.     

基于SDP法诊断发动机的异响

针对发动机异响特征与声信号的复杂性,提出了基于SDP（Symmetrized Dot Pattern）的发动机异响诊断方法。通过将测得的发动机各个局部位置的声信号将其时域波形转换为极坐标图形,利用正常发动机与产生异响发动机SDP图形之间的相关系数来判断所测发动机是否存在异响,与传统诊断方法即对时域与频域信号幅值不同进行对比分析相比具有分析时间短、辨别直观等优点。试验结果证明,SDP法能快速准确地分辨出正常发动机与异响发动机的差别,达到了对发动机异响诊断的目的。     

Integrated design of supply chain networks with three echelons, multiple commodities and technology selection

We consider a strategic supply chain design problem with three echelons, multiple commodities and technology selection. We model the problem as a tri-echelon, capacitated facility location problem that decides on the location of plants and warehouses, their capacity and technology planning, the assignment of commodities to plants and the flow of commodities to warehouses and customer zones. We use a mixed-integer programming formulation strengthened by valid but redundant constraints and apply Lagrangean relaxation to decompose the problem by echelon. Lagrangean relaxation provides a lower bound that is calculated using an interior-point cutting plane method. Feasible solutions are generated using a primal heuristic that uses the solution of the subproblems. Unlike common practice in the literature, the decomposition does not aim at getting easy subproblems, but rather at getting subproblems that preserve most of the characteristics of the original problem. Not only does this provide a sharp lower bound but also leads to a simple and efficient primal heuristic. We can afford to have relatively difficult subproblems because the interior-point cutting plane method used to solve the Lagrangean dual makes clever and selective choices of the Lagrangean multipliers leading to fewer calls to the subproblems. Computational results indicate the efficiency of the approach in providing a sharp bound and in generating feasible solutions that are of high quality.