Solving quadratic convex bilevel programming problems using a smoothing method

In this paper, we present a smoothing sequential quadratic programming to compute a solution of a quadratic convex bilevel programming problem. We use the Karush-Kuhn-Tucker optimality conditions of the lower level problem to obtain a nonsmooth optimization problem known to be a mathematical program with equilibrium constraints; the complementary conditions of the lower level problem are then appended to the upper level objective function with a classical penalty. These complementarity conditions are not relaxed from the constraints and they are reformulated as a system of smooth equations by mean of semismooth equations using Fisher-Burmeister functional. Then, using a quadratic sequential programming method, we solve a series of smooth, regular problems that progressively approximate the nonsmooth problem. Some preliminary computational results are reported, showing that our approach is efficient.     

Convergence analysis and performance of the extended artificial physics optimization algorithm

This paper presents extended artificial physics optimization (EAPO), a population-based, stochastic, evolutionary algorithm (EA) for multidimensional search and optimization. EAPO extends the physicomimetics-based Artificial Physics Optimization (APO) algorithm by including each individual’s best fitness history. Including the history improves EAPO’s search capability compared to APO. EAPO and APO invoke a gravitational metaphor in which the force of gravity may be attractive or repulsive, the aggregate effect of which is to move individuals toward local and global optima. A proof of convergence is presented that reveals the conditions under which EAPO is guaranteed to converge. Discrete-time linear system theory is used to develop a second-order difference equation for an individual’s stochastic position vector as a function of time step. Stable solutions require eigenvalues inside the unit circle, leading to explicit convergence criteria relating the run parameters {m_i, w, G}. EAPO is tested against several benchmark functions with excellent results. The algorithm converges more quickly than APO and with better diversity.     

Function-valued Padé-type approximant via E-algorithm and its applications in solving integral equations

The function-valued Padé-type approximant (FPTA) was defined in the inner product space [8]. In this work, we choose the coefficients in the Neumann power series to make the inner product with both sides a function-valued system of equations to yield a scalar system. Then we express an FPTA in the determinant form. To avoid the direct computation of the determinants, we present the E-algorithm for FPTA based on the vector-valued E-algorithm given by Brezinski [4]. The method of FPTA via E-algorithm (FPTAVEA) not only includes all previous methods but overcomes their essential difficulties. The numerical experiment for a typical integral equation [1] illustrates that the method of FPTAVEA is simpler and more effective for obtaining the characteristic values and the characteristic functions than all previous methods. In addition, this method is also applicable to other Fredholm integral equations of the second kind without explicit characteristic values and characteristic functions. A corresponding example [12] is given and the numerical result is the same as that in [12].     

On solving the forward kinematics of 3RPR planar parallel manipulator using hybrid metaheuristics

Hybrid metaheuristics have been applied with success in solving many real-world problems. This work introduces hybrid metaheuristics to the field of kinematics problem, in particular, for solving the forward kinematics of the 3RPR parallel manipulator. It implements a combination of genetic algorithms and simulated annealing into two popular hybrid metaheuristic techniques. They are combined as teamwork and relay collaborative hybrid metaheuristics and compared to the performance of genetic algorithms and simulated annealing alone. The results show that the meta-heuristic approaches give robust and high quality solutions. Genetic algorithms and teamwork collaborative metaheuristics showed better performance than simulated annealing and relay collaborative metaheuristics. The given metaheuristic methods obtain all the unique solutions and comparisons with algebraic methods show promising results.     

An alternating projected gradient algorithm for nonnegative matrix factorization

Due to the extensive applications of nonnegative matrix factorizations (NMFs) of nonnegative matrices, such as in image processing, text mining, spectral data analysis, speech processing, etc., algorithms for NMF have been studied for years. In this paper, we propose a new algorithm for NMF, which is based on an alternating projected gradient (APG) approach. In particular, no zero entries appear in denominators in our algorithm which implies no breakdown occurs, and even if some zero entries appear in numerators new updates can always be improved in our algorithm. It is shown that the effect of our algorithm is better than that of Lee and Seung’s algorithm when we do numerical experiments on two known facial databases and one iris database.     

Using genetic algorithms for the coordinated scheduling problem of a batching machine and two-stage transportation

This paper considers a coordinated scheduling problem. For the first-stage transportation there is a crane available to transport the product from the warehouse to a batching machine. For the second-stage transportation there is a vehicle available to deliver the completed jobs from the machine shop floor to the customer. The coordinated scheduling problem of production and transportation deals with sequencing the transportation of the jobs and combining them into batches to be processed. The problem of minimizing the sum of the makespan and the total setup cost was proven by Tang and Gong [1] to be strongly NP-hard. This paper proposes two genetic algorithm (GA) approaches for this scheduling problem, with different result representations. The experimental results demonstrate that a regular GA and a modified GA (MGA) can find near-optimal solutions within an acceptable amount of computational time. Among the two proposed metaheuristic approaches, the MGA is superior to the GA both in terms of computing time and the quality of the solution.     

Quantum-inspired space search algorithm (QSSA) for global numerical optimization

This work presents the evolutionary quantum-inspired space search algorithm (QSSA) for solving numerical optimization problems. In the proposed algorithm, the feasible solution space is decomposed into regions in terms of quantum representation. As the search progresses from one generation to the next, the quantum bits evolve gradually to increase the probability of selecting the regions that render good fitness values. Through the inherent probabilistic mechanism, the QSSA initially behaves as a global search algorithm and gradually evolves into a local search algorithm, yielding a good balance between exploration and exploitation. To prevent a premature convergence and to speed up the overall search speed, an overlapping strategy is also proposed. The QSSA is applied to a series of numerical optimization problems. The experiments show that the results obtained by the QSSA are quite competitive compared to those obtained using state-of-the-art IPOP-CMA-ES and QEA.     

Discretization algorithm for fractional order integral by Haar wavelet approximation

A discretization algorithm is proposed by Haar wavelet approximation theory for the fractional order integral. In this paper, the integration time is divided into two parts, one presents the effect of the past sampled data, calculated by the iterative method, and the other presents the effect of the recent sampled data at a fixed time interval, calculated by the Haar wavelet. This method can reduce the amount of the stored data effectively and be applied to the design of discrete-time fractional order PID controllers. Finally, several numerical examples and simulation results are given to illustrate the validity of this discretization algorithm.     

Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-?-asymptotically nonexpansive mappings

We introduce a modified block hybrid projection algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the mixed equilibrium problems. Then, we obtain strong convergence theorems for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Our results extend and improve ones from several earlier works.     

Numerical treatment of solving singular integral equations by using Sinc approximations

In this paper we propose numerical treatment for singular integral equations. The methods are developed by means of the Sinc approximation with smoothing transformations. Such approximation is an effective technique against the singularities of the equations, and achieves exponential convergence. Therefore the methods improve conventional results where only polynomial convergence have been reported. The resulting algebraic system is solved by least squares approximation and leap frog algorithm. Estimation of errors of the approximate solution is presented. Some experimental tests are presented to show the efficient of the proposed methods.