Successive Linear Algorithm Research Articles

A Two-Level Coordination Strategy for Distribution Network Balancing

Uncertain distributed energy resources and uneven load allocation cause the three-phase unbalance in distribution networks (DNs), which may harm the health of power equipment and increase the operational cost. There are emerging opportunities to balance three-phase DNs with a number of power electronic devices installed in the system. In this paper, we propose a novel two-level coordination strategy to improve the network balancing performance, where soft open points (SOPs) and phase switch devices (PSDs) are hierarchically coordinated in the network. At the upper level, a new type of SOPs with the function of phase switching is designed to explore the cross-phase power transfer ability; at the lower level, PSDs are utilized to flexibly allocate individual loads to specific phases. The two-level coordination strategy is typically formulated as a mixed-integer nonlinear programming (MINLP) problem. To solve the model accurately and efficiently, we develop a successive linearization algorithm to approximate it to a mixed-integer linear programming (MILP) problem at each iteration. On this basis, we propose a heuristic time-independent fixing algorithm to further ease the computational burden by eliminating a large number of integer variables in the MILP problem. Numerical simulations are conducted to validate the effectiveness, accuracy, and efficiency of the proposed method.

UAV trajectory optimization using chance-constrained second-order cone programming

It is challenging to generate optimal trajectories for nonlinear dynamic systems under external disturbances. In this brief, we present a novel approach for planning safe trajectories of the chance-constrained trajectory optimization problems with nonconvex constraints. First, the chance constraints are handled by deterministic ones which show its availability. We derive an iterative convex optimization method to solve the optimal control problem. Then the chance-constrained optimal control (CCOCP) problem is reformed to be a nonlinear programming problem (NLP) through the hp-adaptive pseudospectral method. An iterative successive linearization algorithm is detailed to convex the NLP to be a convex optimization one which described as a second-order cone programming problem. We demonstrate the proposed approach on a 3-DoF of unmanned aerial vehicle system under chance-constrained. The simulation results show reliable solutions for the UAV chance-constrained trajectory optimization problem.

D2D-enabled resource management in secrecy-ensured 5G and beyond Heterogeneous networks

Integration of Device-to-Device communication in 5G Heterogeneous networks (HetNets) promises to increase resource utilization and spectrum efficiency (SE). However, the broadcast nature of wireless communication makes it vulnerable to an eavesdropper, who can steal useful information from the transmission channel. Hence, ensuring secure communication for both cellular users (CUs) and D2D pairs is the need of the time. In this paper, we study the optimization problem of resource allocation in 5G and beyond wireless networks to maximize the secrecy rate of both CUs and D2D links from the physical layer security perspective. We consider multiple eavesdroppers overhearing the CUs and D2D pairs. Two optimization problems are formulated: one considering separate channels for CUs and D2D user (DU), and second, considering the shared channel between CU and DU. The nature of the formulated problems is mixed-integer non-linear programming (MINLP) problem and to solve it, we propose an ϵ-optimal based successive linearization algorithm. This algorithm decomposes our original problem into finite sub-problem branches and solves each separately. Simulation results show that the proposed technique can significantly improve the secrecy rate, throughput, and energy efficiency of the network especially in the case of separate channels for CUs and DUs.

A differential game model for closed-loop supply chain participants under carbon emission permits

In the low carbon economy background, supply chain participants need to make their investment and production decisions under the carbon emission consideration. In this study, we propose a closed-loop supply chain (CLSC) network including multiple manufacturing/remanufacturing plants and multiple distribution/collection centers. From a long-term horizon and dynamic perspective, we analyze the optimality conditions of CLSC participants, who compete in a non-cooperative manner under carbon emission permits. We develop a differential game model for the CLSC network based on differential variational inequality, and illustrate that the differential variational inequality is equivalent to the nonlinear complementary problem. A successive linearization algorithm is proposed. In addition, we analyze the impact of subsidy and remanufacturing ratio on the equilibrium results. The results indicate that, under carbon permits, receiving subsidy can motivate plants to input more investment into carbon emission abatement technologies to reduce the carbon emission, which results in the reduction of the total carbon stocks and the increase of the total profits for the plants and the whole CLSC network. In addition, plants should enhance their remanufacturing capability, which brings more profits and a higher sustainability level for the whole supply chain. Our study provides implications for companies in the CLSC to enhance their profits through reducing carbon emissions by increasing technology investment and enhancing remanufacturing capabilities.

Robust recursive absolute value inequalities discriminant analysis with sparseness

In this paper, we propose a novel absolute value inequalities discriminant analysis (AVIDA) criterion for supervised dimensionality reduction. Compared with the conventional linear discriminant analysis (LDA), the main characteristics of our AVIDA are robustness and sparseness. By reformulating the generalized eigenvalue problem in LDA to a related SVM-type “concave-convex” problem based on absolute value inequalities loss, our AVIDA is not only more robust to outliers and noises, but also avoids the SSS problem. Moreover, the additional L1-norm regularization term in the objective makes sure sparse discriminant vectors are obtained. A successive linear algorithm is employed to solve the proposed optimization problem, where a series of linear programs are solved. The superiority of our AVIDA is supported by experimental results on artificial examples as well as benchmark image databases.

Numerical methods for solving applied optimal control problems

For an optimal control problem with state constraints, an iterative solution method is described based on reduction to a finite-dimensional problem, followed by applying a successive linearization algorithm with the use of an augmented Lagrangian. The efficiency of taking into account state constraints in optimal control computation is illustrated by numerically solving several application problems.

COAP 2007 Best Paper Award

Each year, the Computational Optimization and Applications (COAP) editorial board selects a paper from the preceding year’s COAP publications for the Best Paper Award. The recipient of the award for papers published in 2007 is Olvi Mangasarian of the University of Wisconsin, Madison and the University of California, San Diego, for his paper “Absolute Value Programming”, published in Volume 36, pages 43–53. This paper [7] as well as subsequent closely related papers [6, 8, 9] deal with the absolute value equation (AVE) Ax + B|x| = b, where A and B are arbitrary m× n real matrices and |x| denotes the vector with absolute values of the n-dimensional real valued vector x. The significance of this class of NP-hard problems arises partly from the fact that when B = I , the identity matrix, AVE is equivalent to the general linear complementarity problem, 0 ≤ x ⊥Mx + q ≥ 0. Even though problems involving absolute values are NP-hard, they share some very interesting properties with those of linear systems. For example, optimization problems with absolute value constraints possess optimality and duality results similar to those of linear programming, even though the problems are inherently nonconvex. Another interesting property that AVE shares with linear inequalities are theorems of the alternative which are established in this paper. The paper also contains a finite successive linearization algorithm for solving absolute value equations that terminates at a necessary optimality condition. This algorithm has solved all random test problems given to it for which mostly m ≥ 2n or n ≥ 2m, up to size (m,n) of (2000,100) and (100,2000). When m = n and B is invertible, which is the case for the linear complementarity problem formulation, a simpler concave minimization

Absolute value equation solution via concave minimization

The NP-hard absolute value equation (AVE) Ax − |x| = b where \(A\in R^{n\times n}\) and \(b\in R^n\) is solved by a succession of linear programs. The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization. A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations.

Minimum-Support Solutions for Radiotherapy Planning

Computer-generated plans for radiation treatment sometimes involve an unnecessarily large number of gantry angles. Such plans are time consuming and expensive to administer because each distinct angle causes a delay while the gantry is repositioned. To address this issue, we consider an optimization model that generates minimum-support solutions to the treatment planning problem – that is, solutions involving a minimum number of gantry angles. This model is a polyhedral concave program, which is solved using a successive linearization algorithm based on work by Mangasarian [10].

Arbitrary-norm separating plane

A plane separating two point sets in n-dimensional real space is constructed such that it minimizes the sum of arbitrary-norm distances of misclassified points to the plane. In contrast to previous approaches that used surrogates for distance-minimization, the present work is based on a precise norm-dependent explicit closed form for the projection of a point on a plane. This projection is used to formulate the separating-plane problem as a minimization of a convex function on a unit sphere in a norm dual to that of the arbitrary norm used. For the 1-norm, the problem can be solved in polynomial time by solving 2 n linear programs or by solving a bilinear program. For a general p-norm, the minimization problem can be transformed via an exact penalty formulation to minimizing the sum ofa convex function and a bilinear function on a convex set. For the one and infinity norms, a finite successive linearization algorithm can be used for solving the exact penalty formulation.