On a quadratic programming problem involving distances in trees

This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated. Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems.

In this paper, we present a recurrent neural network for solving mixed linear complementarity problems (MLCPs) with positive semi-definite matrices. The proposed neural network is derived based on an NCP function and has a low complexity respect to the other existing models. In theoretical and numerical aspects, global convergence of the proposed neural network is proved. As an application, we show that the proposed neural network can be used to solve linear and convex quadratic programming problems. The validity and transient behavior of the proposed neural network are demonstrated by using five numerical examples.

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.

The optimal control problem of linear quadratic regulation (the LQR problem) for linear discrete-time systems with single input delay is studied in this paper. A new and simple approach is applied to derive the optimal control input sequences. With the established duality, we first show that the LQR problem is equivalent to an optimization problem in Krein space. The latter problem is finally converted to a strictly convex quadratic programming problem. Thus we convert the delayed control input into control input delay-free, and convert a dynamic LQR optimal control problem for the linear discrete-time systems into a static mathematical programming model. The optimal control input sequences are successfully derived by solving this strictly convex quadratic programming problem. Our approach is simple and yet very effective in dealing with the LQR problem for the linear discrete-time systems with single input delay.

Redundant robot manipulators are usually applied to various complex scenarios where harmful noise especially periodic calculating noise always exists. In order to avoid the task failure affected by the periodic noise, a circadian rhythms neural network (CRNN) is applied to solve motion planning problems of redundant robot manipulators suffering from the periodic noise. First, in this article, we formulate a tracking problem of redundant manipulators as a convex quadratic programming (QP) problem. Second, the QP problem is converted into a matrix equation. Third, based on the neural dynamic design method, a CRNN model is exploited and developed to solve the matrix equation of the tracking problem. Comparative simulations between the proposed CRNN and the traditional zeroing neural network show that the CRNN has better robustness and performance on solving the end-effector tracking task. Two physical experiments are conducted to further certify the effectiveness, robustness, and practicability of the proposed CRNN.

Based on the global Krotov successive improvement method, we propose a dual computational algorithm for a discrete optimal control problem corresponding to a convex large-scale quadratic programming problem with a separable functional that arises in the prediction of the direct costs (structural) matrix in dynamic input-output models. With decomposition, we are able to use a special form of the constraint matrix to reduce the problem dimension.

Necessary and sufficient conditions are derived, which are fulfilled by quadratic functions that can be written as a product of two affine-linear functions plus an additive constant. This criterium characterizes convex and nonconvex quadratic programming problems, which can be solved by one ofSwarup's [1966a, 1966b, 1966c] algorithm.

Inference on estimators defined by mathematical programming

The bipartite unconstrained 0–1 quadratic programming problem: Polynomially solvable cases

An optical reconfigurable intelligent surface (ORIS) was expected to offer extra secrecy performance gain in a visible light communication (VLC) system. However, nonlinear impairments involved degrade the confidential signal reception and have not been fully considered in designing physical layer security (PLS). In this paper, a novel PLS approach is proposed for an ORIS-aided VLC system with consideration of practical nonlinear impairments. It is mathematically formulated to be an optimization problem that maximizes the signal-to-interference-plus-distortion-and-noise ratio of the legitimate link, while entirely suppressing that of multiple eavesdroppers by jointly optimizing the beamforming, jamming and clipping at the transmitters, and also the surface configuration in terms of mirror assignments and rotation angles at the ORIS. We decompose this mixed combinatorial and non-convex optimization problem into three sub-problems and elaborately transform them to be conventional convex programming, quadratic programming and nonlinear programming problems, respectively. Moreover, we also develop a time-efficient iterative approach to achieve the suboptimal solution with low-computational complexity. Simulation results demonstrate the improvement of secrecy performance as compared with conventional schemes, and also the robustness to severe nonlinear impairments and spatial correlation, thereby confirming the beneficial insights of this methodology for secure VLC with nonlinear devices.

This paper proposes bounds for the objective function of convex quadratic programming problem (QPP) in general form. The method uses just the eigenvalues of the Hessian matrix. By use of the eigenvalues of the Hessian matrix and solving two simple optimisation problems, we propose an interval which contains the optimal value of QPP. This method can be useful for complicated and large scale optimisation problems as well as for the integer quadratic programming problems and also can be used as a start interval for the other existing mathematical methods for QPP. Another application of the proposed interval is to help the decision maker in real applications to estimate the bounds of the optimal solution. Thus the method is useful from both theoretical and practical approaches. This method can also be applied to solve fractional quadratic programming problems as well as for binary and mixed integer quadratic programming problems. Sensitivity analysis for the objective function is another application of the method which will be discussed. To illustrate the method, several problems are solved.

This paper proposes a locally convexified rigid tube model predictive control for obstacle avoidance for linear, discrete‐time, systems subject to additive, bounded disturbances and state and control constraints. The inherently nonconvex obstacle avoidance constraints are locally convexified and converted into closed polyhedral constraints by deploying recent notions of safe sets and tubes. The proposed method leverages these recent notions of safe sets and tubes to reformulate the original nonconvex optimization problem for obstacle avoidance within the context of disturbances into a computationally highly efficient, strictly convex quadratic programming problem. In order to analyze the separation between a predicted state set and each obstacle avoidance constraint set, we employ the Euclidean distance between these two sets and the radius of the Chebyshev ball of their intersection. In the strict separation case, the separating hyperplanes are constructed by utilizing the separation theorem for convex sets, while in the weak separation case, the construction of separating hyperplanes is achieved through the use of the separation theorem and the properties of convex cones. The proposed method provides guarantees of strong system theoretic properties such as robust recursive feasibility, robust positive invariance and robust stability of the controlled uncertain system. A simulation example is also provided to numerically verify the effectiveness of the proposed method.

For realizing the autonomous dynamic soaring maneuver without relying on known wind shear profiles, this paper develops a real-time trajectory correction approach. The strategy is inspired by the fact that large seabirds do not know the wind profile and yet they presumably can feel the wind variations and predict their own motions. Utilizing only the instant wind information rather than the complete profile, this approach formulates a strictly convex quadratic programming problem based on the instant wind strength increment and the predicted errors to determine bounded control inputs. Initialized by an energy-neutral dynamic soaring trajectory that requires the minimum wind strength, control inputs can be corrected in real time. Simulation results demonstrate the effectiveness of the proposed approach under different unspecified wind profiles.

: Several algorithms for solving problems in linear, quadratic, and nonlinear programming, network flows, and facilities location were developed and analyzed. Results include: (1) The analysis of the computational complexity of the problem of determining an optimally sparse representation of the null space of a matrix, and the development of worst-case bounds for the shadow-vertex simplex algorithm and several heuristics for distance constrained discrete facility location problems and conditional covering problems; (2) The development of efficient algorithms for dense and sparse assignment problems, strictly convex quadratic programming problems, and a nonlinear programming problem that arises when maximizing a correlation coefficient subject to linear constraints; and (3) The application of iterative methods to large sparse equality-constraind quadratic programs and the development of multiple constraint deletion strategies for active-set algorithms for linearly consrained nonlinear programming problems. (Author)