Non-convex Cost Function Research Articles

Variable bin size selection for periestimulus time histograms (PSTH) with minimum mean square error criteria

To date the most common method for extracting neuronal responses is by constructing PSTHs that are time locked to task events. Many parameters of interests such as response magnitude, onset and duration are then calculated from the constructed PSTHs. However the precision of PSTH response estimate critically depends on the choice of bin sizes. This dependence demands objective criteria for bin size selection. A seminal study by Shimazaki and Shinomoto [1] derived an optimal cost function for choosing a fixed bin size for a time varying Poisson process. It is easy to see that using a one-size-fit-all recipe for bin sizes will invariably overestimate and underestimate rate changes for fast and slow fluctuations respectively. Here we extend previous results by calculating the cost function that minimizes mean square error for variable bin sizes with the same assumptions used previously for time varying Poisson processes Cost(N,Δ )=1n2T∑i=1N2ki-(ki-Δik ¯)2Δi. To minimize this nonlinear and nonconvex cost function, we utilize an array of methods some of which are widely used for nonlinear optimization, namely: Active set, Simultaneous perturbation stochastic approximation (SPSA), Genetic Algorithm and an approximate heuristic algorithm. Average performance of each algorithm on a typical simulated neuronal firing is calculated using 50 iterations. All methods resulted in a lower cost function compared to fixed bin size as expected. Plotting the final cost vs the algorithm run time shows that the method of 'Active set' overall has the best cost reduction while still being reasonably fast compared to the fixed bin size approach (Figure ​(Figure1)1) . Further investigation of the properties of this cost function and developing computationally efficient methods for its minimization will be the basis of future work. Figure 1 Comparing the costs and time efforts of the algorithms.

Endogenous business cycles caused by nonconvex costs and interactions

This paper shows that endogenous business cycles (inventory cycles) arise from a combination of nonconvex costs and economic interactions among firms. At the micro level, firm behavior is characterized by lumpiness, and the standard production-smoothing theory is empirically rejected. To account for this, a nonconvex cost function is assumed in our model. It might be expected that even if the microeconomic behavior is lumpy, the effect disappears at the aggregate level because of the law of large numbers. However, we show that if there exist interactions among firms, a regular endogenous cycle emerges at the aggregate level given that the degree of the interaction effect exceeds a critical point. That is, the randomly behaving microeconomic agents generate deterministic collective behavior via interactions. This offers an explanation for the Kitchin cycle.

Exchange market algorithm for economic load dispatch

Abstract This paper presents exchange market algorithm for solving economic load dispatch problems. Exchange market algorithm (EMA) is a new, robust, and strong algorithm to extract the optimal point for global optimization. Inspired by the stock exchange trading method, EMA strives to solve optimization problem. Meticulous investigation of the stock exchange methods employed by the elites in such markets has yielded to shape this algorithm. This algorithm has two searcher operators as well as two absorbent operators for individuals to be absorbed to the elite person, which leads to creation and organization of random numbers in the best way. In order to show the abilities of the EMA, this algorithm has been implemented on four test systems in different dimensions (3, 6, 15 and 40 units) with convex and nonconvex cost functions. The numerical results have been compared with the results of some new and strong algorithms. The results prove the robustness and effectiveness of the proposed algorithm and show that it could be used as a reliable tool for solving practical ELD problems.

MORDUKHOVICH SUBGRADIENTS OF THE VALUE FUNCTION IN A PARAMETRIC OPTIMAL CONTROL PROBLEM

This paper is devoted to the study the first-order behavior of the value function of a parametric optimal control problem with linear constraints and a nonconvex cost function. By establishing an abstract result on the Mordukhovich subdifferential of the value function of a parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the value function to a parametric optimal control problem.

Second-order necessary optimality conditions for a discrete optimal control problem with mixed constraints

In this paper, we study second-order necessary optimality conditions for a discrete optimal control problem with nonconvex cost functions and state-control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order necessary optimality conditions for a discrete optimal control problem.

Nonlinear Model Predictive Control Based on Collective Neurodynamic Optimization

In general, nonlinear model predictive control (NMPC) entails solving a sequential global optimization problem with a nonconvex cost function or constraints. This paper presents a novel collective neurodynamic optimization approach to NMPC without linearization. Utilizing a group of recurrent neural networks (RNNs), the proposed collective neurodynamic optimization approach searches for optimal solutions to global optimization problems by emulating brainstorming. Each RNN is guaranteed to converge to a candidate solution by performing constrained local search. By exchanging information and iteratively improving the starting and restarting points of each RNN using the information of local and global best known solutions in a framework of particle swarm optimization, the group of RNNs is able to reach global optimal solutions to global optimization problems. The essence of the proposed collective neurodynamic optimization approach lies in the integration of capabilities of global search and precise local search. The simulation results of many cases are discussed to substantiate the effectiveness and the characteristics of the proposed approach.

Congestion management by determining optimal location of series FACTS devices using hybrid bacterial foraging and Nelder–Mead algorithm

In this paper, the Thyristor-Controlled Series-Compensated (TCSC) devices are located for congestion management in the power system by considering the non-smooth fuel cost function and penalty cost of emission. For this purpose, it is considered that the objective function of the proposed optimal power flow (OPF) problem is minimizing fuel and emission penalty cost of generators. A hybrid method that is the combination of the bacterial foraging (BF) algorithm with Nelder–Mead (NM) method (BF-NM) is employed to solve the OPF problems. The optimal location of the TCSC devices are then determined for congestion management. The size of the TCSC is obtained by using of the BF-NM algorithm to minimize the cost of generation, cost of emission, and cost of TCSC. The simulation results on IEEE 30-bus, modified IEEE 30-bus and IEEE 118-bus test system confirm the efficiency of the proposed method for finding the optimal location of the TCSC with non-smooth non-convex cost function and emission for congestion management in the power system. In addition, the results clearly show that a better solution can be achieved by using the proposed OPF problem in comparison with other intelligence methods.

Second-Order Necessary Optimality Conditions for a Discrete Optimal Control Problem

In this paper, we study second-order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function and control constraints. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we derive second-order optimality conditions for a discrete optimal control problem.

Solving non-Convex and Restricted Problems Using Swarms – Economic Dispatch Case

Abstract In this paper three heuristic techniques based in swarm intelligence are studied, namely its ability to solve non-convex, non-differentiable and highly restricted optimization problems. The performances of Particle Swarm Optimization, Bee Colony Optimization and Cockroach Swarm Optimization solving a set of benchmark functions will be compared. These techniques will also be applied to an electrical engineering problem, to be precise, the economic dispatch with non-convex cost functions. The results obtained up to now, have been demonstrating that these techniques are able to reach good results in the benchmark functions as well as in the problem of economic dispatch.

Discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res. 93 (2012) 93–105

This short communication presents a discussion of “Continuous quick group search optimizer for solving non-convex economic dispatch problems” by Moradi-Dalvand et al. [Electr. Power Syst. Res.] 93 (2012) 93–105. The discussed paper presented economic dispatch problem by experimenting with five example systems considering 6, 10, 20, 40 and 140-unit test systems considering non-convex cost function with valve point loading effects, multi-fuel effects and prohibited operating zones. However, in the reported results for the 6 and 20-unit test system, the total generation, total loss and the cost quoted were different for the given generation schedule. In this communication, the corrected data of the 6-unit test system and clarification regarding power transmission losses and cost calculations are presented.

Deterministically guided differential evolution for constrained power dispatch with prohibited operating zones

Abstract This paper presents a new approach to solve economic load dispatch (ELD) problem in thermal units with non-convex cost functions using differential evolution technique (DE). In practical ELD problem, the fuel cost function is highly non linear due to inclusion of real time constraints such as valve point loading, prohibited operating zones and network transmission losses. This makes the traditional methods fail in finding the optimum solution. The DE algorithm is an evolutionary algorithm with less stochastic approach to problem solving than classical evolutionary algorithms.DE have the potential of simple in structure, fast convergence property and quality of solution. This paper presents a combination of DE and variable neighborhood search (VNS) to improve the quality of solution and convergence speed. Differential evolution (DE) is first introduced to find the locality of the solution, and then VNS is applied to tune the solution. To validate the DE-VNS method, it is applied to four test systems with non-smooth cost functions. The effectiveness of the DE-VNS over other techniques is shown in general.

On the Gradient Descent Localization of Radioactive Sources

We consider the robust localization of radioactive sources by using their gamma-ray count at the smallest number of sensors needed to theoretically localize. We formulate a class of non-convex cost functions and consider their gradient descent optimization. We show that in N-dimensions, if there are exactly N + 1 sensors and the source lies in their open convex hull, then this convex hull is devoid of false stationary points. Thus we augment gradient descent with random projections into the convex hull, when an estimate leaves it. We argue that convergence in probability to the correct source location, will occur. Simulations demonstrate the efficacy of this algorithm.

Particle swarm optimization with smart inertia factor for solving non-convex economic load dispatch problems

SUMMARY This paper proposes a new approach to solve the non-convex economic dispatch problem using particle swarm optimization with smart inertia factor (PSO-SIF) algorithm in which, inertia coefficients are controlled with respect to cost function in each population. Hence, each population has unique inertia coefficient and as a result unique velocity in convergent direction for the best group solution. The new algorithm has been implemented on four test systems in different dimensions (6, 15, 20 and 40 units) with convex and non-convex cost functions. The numerical results have been compared with the results of a few PSO variants and some recently published results. The results prove the robustness and effectiveness of the proposed method and show that it could be used as a reliable tool for solving optimization problems. Copyright © 2013 John Wiley & Sons, Ltd.

Increasing the reliability of protein–protein interaction networks via non-convex semantic embedding

Over the last decade, the development of high-throughput techniques has resulted in a rapid accumulation of protein–protein interaction (PPI) data. However, the high-throughput experimental interaction data is prone to exhibit high level of noise. Despite the promising performance of current geometric approaches for increasing the reliability of PPI networks, it is still of major concern to find a better method that requires less structural assumptions and is more robust against the large fraction of noisy PPIs. In this paper, we propose a new approach called non-convex semantic embedding (NCSE) for assessing the reliability of interactions. Unlike previous approaches which seek to preserve a predefined distance matrix in the embedding space, NCSE tries to adaptively learn a Euclidean embedding under the simple geometric assumption of PPI networks. We also propose using non-convex cost function in order to improve the robustness of NCSE. The experimental results show that our approach substantially outperforms previous methods on PPI assessment problems. NCSE could thus facilitate further graph-based studies of PPIs and may help infer their hidden underlying biological knowledge. The Matlab source code of NCSE is freely available upon request.

Solving Optimal Power Flow Problems Using Chaotic Self-adaptive Differential Harmony Search Algorithm

Optimal power flow is the basic tool that allows an electric utility to determine the economic and secure operating conditions of an electric power system. This article presents a chaotic self-adaptive differential harmony search algorithm to solve optimal power flow problems with non-smooth and non-convex cost functions. The searching capacity of the proposed chaotic self-adaptive differential harmony search algorithm has been improved by introducing a chaotic self-adaptive differential mutation operator instead of a pitch adjustment operator in the harmony search algorithm. The effectiveness of the proposed chaotic self-adaptive differential harmony search algorithm has been tested with IEEE 30-bus, IEEE 300-bus, and 66-bus Indian utility systems. The simulation results obtained using the proposed algorithm are compared with other variants of the improved harmony search algorithm, such as the differential harmony search algorithm, the chaotic differential harmony search algorithm, the interior point method, and other techniques reported in the literature, to show its effectiveness. The results obtained by the proposed algorithm are found to be better than the results obtained by the differential harmony search, chaotic differential harmony search, interior point method, and other algorithms reported in the literature in terms of solution quality and standard deviation of generation cost. In terms of speed of convergence and computational time, the proposed algorithm is better than the differential harmony search and chaotic differential harmony search algorithms.

Single image deblurring using novel image prior constraints

Single image deblurring is a highly ill-posed problem and requires to be regularized. Many common forms of image prior have a major drawback that is unable to make full use of local image information. In this paper, we propose a single image deblurring method using novel image prior constraints. We establish a probabilistic model by enforcing inspired image prior constraints and adopt an advanced iterative scheme that alternates between blur kernel estimation and non-blind image restoration. To suppress ringing artifacts caused by inevitable blur kernel estimated errors, our method employs total variation image restoration and presents an alternation half-quadratic algorithm to solve the non-convex cost function. Finally, experiments show that our method has good performance in suppressing ringing artifacts, and makes a good balance between alleviating staircase effects and preserving image details.

Emission, reserve and economic load dispatch problem with non-smooth and non-convex cost functions using epsilon-multi-objective genetic algorithm variable

This paper addresses a novel method for the multi-objective economic load dispatch (ELD) problem. Power generation, spinning reserve costs and emission are considered in the objective function of the frequency ELD problem. The frequency deviation, minimum frequency limits and other practical constraints are also taken into account in this problem. It is a highly constrained multi-objective optimization problem that involves conflicting objectives with both equality and inequality constraints. In this paper, an elitist evolutionary multi-objective optimization algorithm based on the concept of ε-dominance, called ε-multi-objective genetic algorithm variable (εv-MOGA), is proposed to solve the frequency ELD problem. In this study, the performance of the proposed εv-MOGA algorithm is compared with the performance of other classic and intelligent algorithms. The proposed method is tested on 6, 10, 13 and 40 generating units, and the simulation results of four power systems demonstrate the advantages of the proposed method for reducing the cost function.

Mordukhovich subgradients of the value function to a parametric discrete optimal control problem

This paper is devoted to the study of the first-order behavior of the value function of a parametric discrete optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Mordukhovich subdifferential of the value function of a parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the value function to a parametric discrete optimal control problem.

T-LSE: a novel robust geometric approach for modeling protein-protein interaction networks.

Protein-protein interaction (PPI) networks provide insights into understanding of biological processes, function and the underlying complex evolutionary mechanisms of the cell. Modeling PPI network is an important and fundamental problem in system biology, where it is still of major concern to find a better fitting model that requires less structural assumptions and is more robust against the large fraction of noisy PPIs. In this paper, we propose a new approach called t-logistic semantic embedding (t-LSE) to model PPI networks. t-LSE tries to adaptively learn a metric embedding under the simple geometric assumption of PPI networks, and a non-convex cost function was adopted to deal with the noise in PPI networks. The experimental results show the superiority of the fit of t-LSE over other network models to PPI data. Furthermore, the robust loss function adopted here leads to big improvements for dealing with the noise in PPI network. The proposed model could thus facilitate further graph-based studies of PPIs and may help infer the hidden underlying biological knowledge. The Matlab code implementing the proposed method is freely available from the web site: http://home.ustc.edu.cn/~yzh33108/PPIModel.htm.

Discussion on “A GA-API Solution for the Economic Dispatch of Generation in Power System Operation”

The commentors feel the authors of the above-named article [ibid., vol. 27, no. 1, pp. 233-242, Feb. 2012] that presents a hybrid heuristic method combining real coded genetic algorithm and special class of ant colony optimization for economic dispatch (ED) problems is commendable. The feasibility of the proposed method to solve ED is shown, by experimenting with four test systems considering 3, 6, 15, and 40 units with convex and nonconvex cost functions and also three benchmark functions. However, we would like to seek the authors' clarification regarding four particular points.