On the scalability of truss geometry and topology optimization with global stability constraints via chordal decomposition

In this paper, we introduce geometry optimization into an existing topology optimization workflow for truss structures with global stability constraints, assuming a linear buckling analysis. The design variables are the cross-sectional areas of the bars and the coordinates of the joints. This makes the optimization problem formulations highly nonlinear and yields nonconvex semidefinite programming problems, for which there are limited available numerical solvers compared with other classes of optimization problems. We present problem instances of truss geometry and topology optimization with global stability constraints solved using a standard primal-dual interior point implementation. During the solution process, both the cross-sectional areas of the bars and the coordinates of the joints are concurrently optimized. Additionally, we apply adaptive optimization techniques to allow the joints to navigate larger move limits and to improve the quality of the optimal designs.

We study the lift-and-project procedures of Lovász and Schrijver for 0-1 integer programming problems. We prove that the procedure using the positive semidefiniteness constraint is not better than the one without it, in the worst case. Various examples are considered. We also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help.

The robust adaptive beamforming problem for general-rank signal model with positive semi-definite (PSD) constraint is considered. The existing approaches for solving the corresponding non-convex optimization problem are iterative methods for which the convergence is not guaranteed. Moreover, these methods solve the problem only suboptimally. We revisit this problem and develop a new beamforming method based on a new solution for the corresponding optimization problem. The new proposed method is iterative and is based on a reformulation and then linearization of a single non-convex difference-of-two-convex functions (DC) constraint. Our simulation results confirm that the new proposed method finds the global optimum of the problem in few iterations and outperforms the state-of-the-art robust adaptive beamforming methods for general-rank signal model with PSD constraint.

The article presents the technique for simultaneous topology, shape and sizing optimisation of tall guyed masts under wind loadings and self-weight using simulated annealing. The objective function is the mass of the mast structure including guys, while the set of design parameters may consist of up to 10 parameters of different nature. The constraints are assessed according to Eurocodes and include the local and global stability constraints, limitations on the slenderness in mast elements, and strength constraints. The proposed optimisation technique covers three independent parts: the optimisation algorithm, meshing pre-processor that yields computational scheme of mast depending on the design parameters, and finite element program evaluating the objective function and constraints via penalisation technique. As an example the results of optimisation of a typical 60 m tall guyed telecommunication mast with different antenna areas are presented. On the basis of these results, the authors try ascertaining the approximate optimal diapasons of geometry and topology parameters such as the width of the shaft, distance of the guy foundation from the mast axis, heights of the guy attachment levels and so on. The authors hope, this will be helpful for constructors as an initial design of mast topology, shape and element sizing.

This paper deals with the frame topology optimization under the frequency constraint and proposes an algorithm that solves a sequence of relaxation problems to obtain a local optimal solution with high quality. It is known that an optimal solution of this problem often has multiple eigenvalues and the feasible set is disconnected. Due to these two difficulties, conventional nonlinear programming approaches often converge to a local optimal solution that is unacceptable from a practical point of view. In this paper, we formulate the frequency constraint as a positive semidefinite constraint of a certain symmetric matrix, and then relax this constraint to make the feasible set connected. The proposed algorithm solves a sequence of the relaxation problems with gradually decreasing the relaxation parameter. The positive semidefinite constraint is treated with the logarithmic barrier function and, hence, the algorithm finds no difficulty in multiple eigenvalues of a solution. Numerical experiments show that global optimal solutions, or at least local optimal solutions with high qualities, can be obtained with the proposed algorithm.

The positive semidefinite constraint for the variable matrix in semidefinite programming (SDP) relaxation is further relaxed by a finite number of second order cone constraints in second order cone programming (SOCP) relaxations. A few types of SOCP relaxations are obtained from different ways of expressing the positive semidefinite constraint of the SDP relaxation. We present how such SOCP relaxations can be derived, and show the relationship between the resulting SOCP relaxations.

Truss layout optimization problems with global stability constraints are nonlinear and nonconvex and hence very challenging to solve, particularly when problems become large. In this paper, a relaxation of the nonlinear problem is modelled as a (linear) semidefinite programming problem for which we describe an efficient primal-dual interior point method capable of solving problems of a scale that would be prohibitively expensive to solve using standard methods. The proposed method exploits the sparse structure and low-rank property of the stiffness matrices involved, greatly reducing the computational effort required to process the associated linear systems. Moreover, an adaptive ‘member adding’ technique is employed which involves solving a sequence of much smaller problems, with the process ultimately converging on the solution for the original problem. Finally, a warm-start strategy is used when successive problems display sufficient similarity, leading to fewer interior point iterations being required. We perform several numerical experiments to show the efficiency of the method and discuss the status of the solutions obtained.

Switched reluctance machines (SRMs) have recently attracted more interest in many applications due to the volatile prices of rare-earth permanent magnets (PMs) used in permanent magnet synchronous machines (PMSMs). They also have rugged construction and can operate at high speeds and high temperatures. However, acoustic noise and high torque ripples, in addition to the relatively low torque density, present significant challenges. Geometry and topology optimization are applied to overcome these challenges and enable SRMs to compete with PMSMs. Key geometric design parameters are optimized to minimize various objective functions within geometry optimization. On the other hand, the material distribution in a particular design space within the machine domain may be optimized using topology optimization. We discuss how these techniques are applied to optimize the geometries and topologies of SRMs to enhance machine performance. As optimizing the machine geometry and material distribution at the design phase is of substantial significance, this work offers a comprehensive literature review on the current state of the art and the possible trends in the optimization techniques of SRMs. The paper also reviews different configurations of SRMs and stochastic and deterministic optimization techniques utilized in optimizing different configurations of the machine.

In this paper, we propose a new method to solve the robust adaptive beamforming problem for the general rank signal model with positive semi-definite constraint. On applying the worst-case performance optimization approach, the considered robust adaptive beamforming problem generates a non-convex optimization problem. Here, we propose a two step closed form solution of the formulated problem, wherein a new single variable minimization problem is constructed. Result of this minimization is used to solve the robust adaptive beamforming problem. Simulation results verify the improvement in the performance by the proposed method over the current robust adaptive beamforming methods for general-rank signal model.

We address the performance optimization problem in a single-station multiclass queueing network with changeover times by means of the achievable region approach. This approach seeks to obtain performance bounds and scheduling policies from the solution of a mathematical program over a relaxation of the system's performance region. Relaxed formulations (including linear, convex, nonconvex and positive semidefinite constraints) of this region are developed by formulating equilibrium relations satisfied by the system, with the help of Palm calculus. Our contributions include: (1) new constraints formulating equilibrium relations on server dynamics; (2) a flow conservation interpretation of the constraints previously derived by the potential function method; (3) new positive semidefinite constraints; (4) new work decomposition laws for single-station multiclass queueing networks, which yield new convex constraints; (5) a unified buffer occupancy method of performance analysis obtained from the constraints; (6) heuristic scheduling policies from the solution of the relaxations.

Singular Value Decomposition (SVD) is a well studied research topic in many fields and applications from data mining to image processing. Data arising from these applications can be represented as a matrix where this matrix is large and sparse. Most existing algorithms are used to calculate singular values, left and right singular vectors of a largedense matrix but not large-sparse matrix. Even if they can find SVD of a large matrix, calculation of large-dense matrix has high time complexity due to sequential algorithms. Distributed approaches are proposed for computing SVD of large matrices. However, rank of the matrix is still being a problem when solving SVD with these distributed algorithms. In this paper we propose Ranky, set of methods to solve rank problem on large-sparse matrices in a distributed manner. Experimental results show that the Ranky approach recovers singular values, singular left and right vectors of a given large-sparse matrix with negligible error.

An integer programming approach for solving the p-dispersion problem

Fog computing is a paradigm that extends cloud computing services to the edge of the network in order to support delay-sensitive Internet of Things (IoT) services. One of the most promising use-cases of fog computing is Smart City scenarios. Fog computing can substantially improve the quality of citywide services by reducing response delays. Owing to geographically distributed and resource-constrained fog nodes and a multitude of IoT devices in Smart Cities, efficient service deployment and end device traffic routing are quite challenging. Therefore, in this paper, we present an Integer Linear Programming (ILP) formulation for the Joint Application Component Placement and Traffic Routing (JAcPTR) problem in which users’ delay requirements and the limited traffic processing capacity of application instances are considered. Besides, the JAcPTR enables users and infrastructure managers to easily enforce their locality and management requirements in the deployment of application instances. To cope with the considerably high execution time in large instances of the JAcPTR problem, we propose a fast polynomial-time heuristic to efficiently solve the problem. The performance of the proposed heuristic has been evaluated through extensive simulation. Results show that in large instances of the problem, while the state-of-the-art Mixed Integer Linear Programming (MILP) solver fails to obtain a solution in 50% of the simulation runs in 300 seconds, our proposed heuristic can obtain a near-optimal solution in less than one second.

Distance metric learning aims to learn from the given training data a valid distance metric, with which the similarity between data samples can be more effectively evaluated for classification. Metric learning is often formulated as a convex or nonconvex optimization problem, while most existing methods are based on customized optimizers and become inefficient for large scale problems. In this paper, we formulate metric learning as a kernel classification problem with the positive semi-definite constraint, and solve it by iterated training of support vector machines (SVMs). The new formulation is easy to implement and efficient in training with the off-the-shelf SVM solvers. Two novel metric learning models, namely positive-semidefinite constrained metric learning (PCML) and nonnegative-coefficient constrained metric learning (NCML), are developed. Both PCML and NCML can guarantee the global optimality of their solutions. Experiments are conducted on general classification, face verification, and person re-identification to evaluate our methods. Compared with the state-of-the-art approaches, our methods can achieve comparable classification accuracy and are efficient in training.

An efficient variable neighborhood search heuristic for very large scale vehicle routing problems