Modified Subgradient Algorithm Research Articles

A HYBRID MODIFIED SUBGRADIENT ALGORITHM THAT SELF-DETERMINES THE PROPER PARAMETER VALUES

A successful solution algorithm for non-convex optimization problems is the Modified Subgradient Algorithm (MSGA), which solves dual problems based on the sharp augmented lagrangian function. However, its performance highly depends on its parameter values, and determining the appropriate parameter values is difficult as they can be completely different for each problem. In this study, a new hybrid solution approach that a tabu search algorithm to find the appropriate MSGA parameter values and the MSGA algorithm run together is proposed. Although it seems like a contradiction to use an algorithm that also has its parameters to determine the most appropriate parameter values of an algorithm, this contradiction is eliminated by fixing the parameter values of the tabu search algorithm. The proposed algorithm does not need appropriate values of any algorithm parameter. It can find appropriate parameter values for each problem itself starting with the same fixed initial values. To show the success of the developed algorithm, especially on 0-1 quadratic problems, it is compared with the classical MSGA algorithm by using the quadratic knapsack test instances taken in the literature. According to the obtained solutions, the superiority of the hybrid algorithm has been demonstrated.

Application of Modified Subgradient Algorithm Based on Feasible Values to Security Constrained Non-Convex Economic Dispatch Problems with Prohibited Zones and Ramp Rates

Application of Modified Subgradient Algorithm Based on Feasible Values to Security Constrained Non-Convex Economic Dispatch Problems with Prohibited Zones and Ramp Rates

Solution to Security-constrained Non-convex Pumped-Storage Hydraulic Unit Scheduling Problem by Modified Subgradient Algorithm Based on Feasible Values and Pseudo Water Price

A security-constrained power dispatch problem with a non-convex cost function for a lossy electric power system area including a pumped-storage hydraulic unit is formulated. Then, an iterative solution method based on a modified subgradient algorithm operating on feasible values and pseudo water price for the pumped-storage hydraulic unit is used to solve it. In the iterative proposed solution method, the modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the pseudo water price for the pumped-storage hydraulic unit is employed to adjust the net amount of water used by the pumped-storage hydraulic unit during the considered operation period. Since all equality and inequality constraints in the proposed non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles, the off-nominal tap settings, and susceptances values of SVAR systems, they are taken as independent variables. Load flow equations are added into the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, off-nominal tap setting constraints, and SVAR system susceptance value constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method, which does not add any extra independent variable into the model, before its application to the optimization model. Since the method does not add any extra independent variable into the model, the solution time is reduced further. The proposed dispatch technique is tested on a sample power system that has 12 buses with 5 thermal units and a pumped-storage hydraulic unit. Optimal total cost value for the power system without any pumped-storage unit is calculated first. Later, the same optimal total cost value for the power system with the pumped-storage unit is recalculated, and the obtained saving in the optimal total cost value, due to the employment of the pumped-storage unit, is presented. The solution times for the dispatch problems with and without the pumped-storage unit is also presented. At the end, the modified subgradient algorithm based on feasible values is applied directly to the dispatch problem with a pumped-storage unit. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem with the pumped-storage unit.

Determining the parameters of MSG algorithm for multi period layout problem

PurposeThe facility layout problem aims to assign machines/departments to locations and modeled as a quadratic assignment problem (QAP). Multi period facility layout is a special case of this problem where the sum of material handling and re‐layout costs are minimized. Since the problem is proved to be NP‐hard, several exact and heuristic methods are proposed in the literature. The purpose of this paper is to solve the multi period layout problem by using the modified sub‐gradient (MSG) algorithm for the first time and to determine its parameters.Design/methodology/approachThe MSG algorithm can solve a large‐scale of optimization problems that also includes multi period facility layout. Since the performance of the algorithm depends on parameters, a design of experiment is made to determine the appropriate parameter values.FindingsThe proposed method evaluates the parameters of the MSG algorithm and most suitable general algebraic modeling solvers. It is observed that the parameter α value and solver type have main effects for small and large size test problems. Further, the results stated that solver type has more influence on large size test problem.Research limitations/implicationsThis study is limited with the determination of the MSG algorithm parameters and solver types on the well known small and large size test problems. Further studies may include other test problem results obtained from the presented MSG algorithm parameters and compare them with best known results in the literature.Originality/valueThe paper determines the parameters of the MSG algorithm that is used to solve the multi period layout problem, for the first time in the literature.

A genetic algorithm extended modified sub-gradient algorithm for cell formation problem with alternative routings

This paper addresses the cell formation problem with alternative part routes. The problem is considered in the aspect of the natural constraints of real-life production systems such as cell size, separation and co-location constraints. Co-location constraints were added to the proposed model in order to deal with the necessity of grouping certain machines in the same cell for technical reasons, and separation constraints were included to prevent placing certain machines in close vicinity. The objective is to minimise the weighted sum of the voids and the exceptional elements. A hybrid algorithm is proposed to solve this problem. The proposed algorithm hybridises the modified sub-gradient (MSG) algorithm with a genetic algorithm. MSG algorithm solves the sharp augmented Lagrangian dual problems, where zero duality gap property is guaranteed for a wide class of optimisation problems without convexity assumption. Generally, the dual problem is solved by using GAMS solvers in the literature. In this study, a genetic algorithm has been used for solving the dual problem at the first time. The experimental results show the advantage of combining the MSG algorithm and the genetic algorithm. Although the MSG algorithm, whose dual problem is solved by GAMS solver, and the genetic algorithm cannot find feasible solutions, hybrid algorithm generates feasible solutions for all of the test problems.

A Security-constrained Economic Power Dispatch Technique Using Modified Subgradient Algorithm Based on Feasible Values and Pseudo Scaling Factor for a Power System Area Including Limited Energy Supply Thermal Units

A security-constrained power dispatch problem with non-convex cost function for a lossy electric power system area including limited energy supply thermal units fueled under a take-or-pay agreement is formulated. Then, an iterative solution method based on a modified subgradient algorithm based on feasible values and common pseudo scaling factor for limited energy supply thermal units is proposed and used to solve the problem. In the iterative proposed solution method, a modified subgradient algorithm based on feasible values is used to solve the dispatch problem in each subinterval, while the common pseudo scaling factor for limited energy supply thermal units is employed to adjust the amount of fuel consumed by the limited energy supply units during the considered operation period. Since all equality and inequality constraints in the non-linear optimization model of a subinterval are functions of bus voltage magnitudes and phase angles (the off-nominal tap settings), they are taken as independent variables except for those belonging to the reference bus. Load flow equations are added to the model as equality constraints. The unit generation constraints, transmission line capacity constraints, bus voltage magnitude constraints, and off-nominal tap setting constraints are added into the optimization problem as inequality constraints. Since the modified subgradient algorithm based on feasible values requires that all inequality constraints should be expressed in equality constraint form, all inequality constraints are converted into equality constraints by the method given in this article before application of the modified subgradient algorithm based on feasible values to the optimization model. The proposed technique is tested on an IEEE 30-bus test system with a non-convex total cost rate curve. A portion of the dispatch problem considered in this article, where only a loading of the system in a subinterval is considered, was also solved by some other recently reported dispatch techniques. First, minimum total fuel cost rates and solution times obtained from the modified subgradient algorithm based on feasible values versus other techniques are compared, and the outperformance of the modified subgradient algorithm based on feasible values in terms of both total cost rate and solution time is demonstrated. After that, the optimal total fuel cost value and solution time of the system for the operation period for the cases where the fuel constraint is ignored and not ignored are presented, and it is shown that adding the fuel constraint into the dispatch problem further decreases the optimal total fuel cost for the considered dispatch problem. Finally, the modified subgradient algorithm based on feasible values was applied to the fuel constrained dispatch problem directly. It is demonstrated that the proposed solution method, where the modified subgradient algorithm based on feasible values is employed to solve an interval's dispatch problem, gives a solution time that is smaller than that obtained from the direct application of the modified subgradient algorithm based on feasible values to the dispatch problem.

The modified subgradient algorithm based on feasible values

In this article, we continue to study the modified subgradient (MSG) algorithm previously suggested by Gasimov for solving the sharp augmented Lagrangian dual problems. The most important features of this algorithm are those that guarantees a global optimum for a wide class of non-convex optimization problems, generates a strictly increasing sequence of dual values, a property which is not shared by the other subgradient methods and guarantees convergence. The main drawbacks of MSG algorithm, which are typical for many subgradient algorithms, are those that uses an unconstrained global minimum of the augmented Lagrangian function and requires knowing an approximate upper bound of the initial problem to update stepsize parameters. In this study we introduce a new algorithm based on the so-called feasible values and give convergence theorems. The new algorithm does not require to know the optimal value initially and seeks it iteratively beginning with an arbitrary number. It is not necessary to find a global minimum of the augmented Lagrangian for updating the stepsize parameters in the new algorithm. A collection of test problems are used to demonstrate the performance of the new algorithm.

A primal dual modified subgradient algorithm with sharp Lagrangian

We apply a modified subgradient algorithm (MSG) for solving the dual of a nonlinear and nonconvex optimization problem. The dual scheme we consider uses the sharp augmented Lagrangian. A desirable feature of this method is primal convergence, which means that every accumulation point of a primal sequence (which is automatically generated during the process), is a primal solution. This feature is not true in general for available variants of MSG. We propose here two new variants of MSG which enjoy both primal and dual convergence, as long as the dual optimal set is nonempty. These variants have a very simple choice for the stepsizes. Moreover, we also establish primal convergence when the dual optimal set is empty. Finally, our second variant of MSG converges in a finite number of steps.

The Performance of the Modified Subgradient Algorithm on Solving the 0–1 Quadratic Knapsack Problem

In this study, the performance of the modified subgradient algorithm (MSG) to solve the 0-1 quadratic knapsack problem (QKP) was examined. The MSG was proposed by Gasimov for solving dual problems constructed with respect to sharp Augmented Lagrangian function. The MSG has some important proven properties. For example, it is convergent, and it guarantees duality for the problems such that its objective and constraint functions are all Lipschtz. Additionally, the MSG has been successfully used for solving non-convex continuous and some combinatorial problems with equality constraints since it was first proposed. In this study, the MSG was used to solve the QKP which has an inequality constraint. The first step in solving the problem was converting zero-one nonlinear QKP problem into continuous nonlinear problem by adding only one constraint and not adding any new variables. Second, in order to solve the continuous QKP, dual problem with zero duality gap was constructed by using the sharp Augmented Lagrangian function. Finally, the MSG was used to solve the dual problem, by considering the equality constraint in the computation of the norm. To compare the performance of the MSG with some other methods, some test instances from the relevant literature were solved both by using the MSG and by using three different MINLP solvers of GAMS software. The results obtained were presented and discussed.

An inexact modified subgradient algorithm for nonconvex optimization

We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may allow to solve problems with less computational effort. We illustrate this through test problems, including an optimal bang-bang control problem, under several different inexactness schemes.