Quadratic Programming Subproblem Research Articles

A projected-search interior-point method for nonlinearly constrained optimization

AbstractThis paper concerns the formulation and analysis of a new interior-point method for constrained optimization that combines a shifted primal-dual interior-point method with a projected-search method for bound-constrained optimization. The method involves the computation of an approximate Newton direction for a primal-dual penalty-barrier function that incorporates shifts on both the primal and dual variables. Shifts on the dual variables allow the method to be safely “warm started” from a good approximate solution and avoids the possibility of very large solutions of the associated path-following equations. The approximate Newton direction is used in conjunction with a new projected-search line-search algorithm that employs a flexible non-monotone quasi-Armijo line search for the minimization of each penalty-barrier function. Numerical results are presented for a large set of constrained optimization problems. For comparison purposes, results are also given for two primal-dual interior-point methods that do not use projection. The first is a method that shifts both the primal and dual variables. The second is a method that involves shifts on the primal variables only. The results show that the use of both primal and dual shifts in conjunction with projection gives a method that is more robust and requires significantly fewer iterations. In particular, the number of times that the search direction must be computed is substantially reduced. Results from a set of quadratic programming test problems indicate that the method is particularly well-suited to solving the quadratic programming subproblem in a sequential quadratic programming method for nonlinear optimization.

An Inexact Sequential Quadratic Programming Method for Learning and Control of Recurrent Neural Networks.

This article considers the two-stage approach to solving a partially observable Markov decision process (POMDP): the identification stage and the (optimal) control stage. We present an inexact sequential quadratic programming framework for recurrent neural network learning (iSQPRL) for solving the identification stage of the POMDP, in which the true system is approximated by a recurrent neural network (RNN) with dynamically consistent overshooting (DCRNN). We formulate the learning problem as a constrained optimization problem and study the quadratic programming (QP) subproblem with a convergence analysis under a restarted Krylov-subspace iterative scheme that implicitly exploits the structure of the associated Karush-Kuhn-Tucker (KKT) subsystem. In the control stage, where a feedforward neural network (FNN) controller is designed on top of the RNN model, we adapt a generalized Gauss-Newton (GGN) algorithm that exploits useful approximations to the curvature terms of the training data and selects its mini-batch step size using a known property of some regularization function. Simulation results are provided to demonstrate the effectiveness of our approach.

Structure-exploiting Newton-type method for optimal control of switched systems

ABSTRACT This study proposes an efficient Newton-type method for the optimal control of switched systems under a given mode sequence. A mesh-refinement-based approach is utilised to discretise continuous-time optimal control problems (OCPs) and formulate a nonlinear program (NLP), which guarantees the local convergence of a Newton-type method. A dedicated structure-exploiting algorithm (Riccati recursion) is proposed to perform a Newton-type method for the NLP efficiently because its sparsity structure is different from a standard OCP. The proposed method computes each Newton step with linear time-complexity for the total number of discretization grids as the standard Riccati recursion algorithm. Additionally, the computation is always successful if the solution is sufficiently close to a local minimum. Conversely, general quadratic programming (QP) solvers cannot accomplish this because the Hessian matrix is inherently indefinite. Moreover, a modification on the reduced Hessian matrix is proposed using the nature of the Riccati recursion algorithm as the dynamic programming for a QP subproblem to enhance the convergence. A numerical comparison is conducted with off-the-shelf NLP solvers, which demonstrates that the proposed method is up to two orders of magnitude faster. Whole-body optimal control of quadrupedal gaits is also demonstrated and shows that the proposed method can achieve the whole-body model predictive control (MPC) of robotic systems with rigid contacts.

Energy-efficient predictive control for trams incorporating disjunctive time constraints from traffic lights

Tram operations are often blocked by traffic lights, leading to frequent decelerations and re-accelerations that increase operational energy consumption. This paper focuses on tram energy-efficient control problem incorporating time constraints from traffic lights that have multiple feasible green time windows (GTWs). We formulate the problem as a mixed-integer nonlinear program (MINLP), where binary variables are assigned to model disjunctive time constraints of the GTWs. To address computational challenge of solving the MINLP, we reformulate it as a tractable nonlinear program (NLP). Specifically, an equivalent NLP is first presented by replacing the integrality constraint with nonlinear constraints, and then the nonlinear constraints are relaxed and penalized into cost functions. To recover a solution of the MINLP, we propose a computationally efficient sequential quadratic programming algorithm in a shrinking horizon model predictive control framework, which updates the penalty parameter and quadratic programming subproblems in parallel. The solution obtained from the subproblem is feasible in each iteration, and convergence of the feasibility iterations can be enforced by the updated penalty. The performance of the proposed approach is investigated on different scenarios using real-life tram data. Results show that the method is able to generate energy-efficient driving trajectories in a dynamic environment, while crossing traffic lights in effective GTWs without unnecessary decelerations and re-accelerations.

Interference optimized dual-functional radar-communication waveform design with low PAPR and range sidelobe

Dual-functional radar-communication system integrates the radar and communication functions into one system with a shared dual-functional waveform. In this paper, we focus on the problem of dual-functional waveform design under the peak-to-average power ratio (PAPR) constraint and low range sidelobe requirement. When the conventional schemes aim to minimize the multi-user interference (MUI) energy and maximize the similarity to the reference radar waveform, in our proposed method, a weighted summation objective function incorporated with the PAPR constraint and optimized MUI is constructed for the waveform design. Specifically, MUI is classified into destructive interference and constructive interference, while the latter is exploited as a free power for communication and is constrained to guarantee the communication quality-of-service (QoS). To tackle the resulting high order nonconvex problem, we develop an iteration algorithm based on the alternating direction method of multipliers (ADMM) procedure, which involves two quadratic programming subproblems in each iteration and is easy to solve with the current solver. Moreover, by taking the imperfect channel state information (CSI) into consideration, we further investigate the robust dual-functional waveform design problem with the bounded uncertainty of the CSI. Numerical simulations validate the effectiveness of the proposed method.

A stabilized sequential quadratic semidefinite programming method for degenerate nonlinear semidefinite programs

In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving degenerate nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Unlike the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems inexactly, and each QSDP is feasible. One more remarkable point of the proposed method is that constraint qualifications (CQs) or boundedness of Lagrange multiplier sequences are not required in the global convergence analysis. Specifically, without assuming such conditions, we prove the global convergence to a point satisfying any of the following: the stationary conditions for the feasibility problem, the approximate-Karush-Kuhn-Tucker (AKKT) conditions, and the trace-AKKT conditions. Finally, we conduct some numerical experiments to examine the efficiency of the proposed method.

Joint Communication and Computation Resource Scheduling of a UAV-Assisted Mobile Edge Computing System for Platooning Vehicles

Connected and autonomous vehicles (CAVs) are recently envisioned to provide a tremendous social impact, while they put forward a much higher requirement for both vehicular communication and computation capacities to process resource-intensive applications. In this paper, we study unmanned aerial vehicle (UAV)-assisted mobile edge computing (MEC) for a platoon of wireless power transmission (WPT)-enabled vehicles. Our objective is to maximize the system-wide computation capacity under both communication and computation resource constraints. We incorporate the coupled effects of the platooning vehicles and the flying UAV, air-to-ground (A2G) and ground-to-air (G2A) communications, onboard computing and energy harvesting into a joint scheduling optimization model of communication and computation resources. To tackle the resulting optimization problem, we propose a successive convex programming method based on a second-order convex approximation, in which feasible search directions are obtained by solving a sequence of quadratic programming subproblems and used to generate feasible points that can approach a local optimum. We also theoretically prove the feasibility and convergence of the proposed method. Moreover, simulation results are provided to validate the effectiveness of our proposed method and demonstrate its superior performance over other conventional schemes.

An SQP Algorithm for Structural Topology Optimization Based on Majorization–Minimization Method

When applying the sequential quadratic programming (SQP) algorithm to topology optimization, using the quasi-Newton methods or calculating the Hessian matrix directly will result in a considerable amount of calculation, making it computationally infeasible when the number of optimization variables is large. To solve the above problems, this paper creatively proposes a method for calculating the approximate Hessian matrix for structural topology optimization with minimum compliance problems. Then, the second-order Taylor expansion transforms the original problem into a series of separable and easy-to-solve convex quadratic programming (QP) subproblems. Finally, the quadratic programming optimality criteria (QPOC) method and the QP solver of MATLAB are used to solve the subproblems. Compared with other sequential quadratic programming methods, the advantage of the proposed method is that the Hessian matrix is diagonally positive definite and its calculation is simple. Numerical experiments on an MBB beam and cantilever beam verify the feasibility and efficiency of the proposed method.

Multidisciplinary Design Optimization Approach to Integrated Space Mission Planning and Spacecraft Design

Space mission planning and spacecraft design are tightly coupled and need to be considered together for optimal performance; however, this integrated optimization problem results in a large-scale mixed-integer nonlinear programming (MINLP) problem, which is challenging to solve. In response to this challenge, this paper proposes a new solution approach to this problem based on decomposition-based optimization via augmented Lagrangian coordination. The proposed approach leverages the unique structure of the problem that enables its decomposition into a set of coupled subproblems of different types: a mixed-integer quadratic programming (MIQP) subproblem for mission planning, and one or more nonlinear programming (NLP) subproblem(s) for spacecraft design. Because specialized MIQP or NLP solvers can be applied to each subproblem, the proposed approach can efficiently solve the otherwise intractable integrated MINLP problem. An automatic and effective method to find an initial solution for this iterative approach is also proposed so that the optimization can be performed without a user-defined initial guess. The demonstration case study shows that, compared to the state-of-the-art method, the proposed formulation converges substantially faster and the converged solution is at least the same or better given the same computational time limit.

Spectrally compatible multiple‐input multiple‐output radar waveform design based on alternating direction method of multipliers

Multiple-input multiple-output (MIMO) radar waveform design in a spectrally crowded environment is a challenging problem. In this study, the issue of MIMO radar waveform design in coexistence with communication systems is investigated by maximising the output signal-to-interference-plus-noise ratio (SINR) of the MIMO radar under transmit energy, similarity and spectral compatibility constraints. Different from the existing alternating direction method of multipliers (ADMM) proposed in a single-antenna scenario, a novel ADMM algorithm for spectrally compatible MIMO radar waveform design is proposed to efficiently tackle the non-convex optimisation problem in both single-antenna and multi-antenna scenarios. Due to the decomposability of ADMM, only a few quadratic programming subproblems need to be solved during each iteration in the authors' algorithm, which reduces the computational complexity. In numerical simulations, the proposed ADMM algorithm is compared with the semidefinite relaxation algorithm and the existing ADMM algorithm in a single-antenna scenario and a multi-antenna scenario, respectively. It shows that the proposed ADMM algorithm can achieve almost the same performance as the two existing algorithms; however, it is computationally more efficient.

Relay beamforming design for physical layer secure communication via line search algorithm

This paper considers a physical layer security model in wireless communications. Two legitimate users communicate through several relays with the presence of an eavesdropper. We jointly design the relay beamforming weights and minimize the total relay transmit power, while ensuring users' Quality of Services and preventing the information being eavesdropped at the same time. The problem is a robust optimization problem, because of the imperfect channel state information from users and relays to the eavesdropper. First the original problem is simplified, where the high order robust terms are omitted. Then we design an iterative algorithm based on line search, by solving two Quadratically Constrained Quadratic Programming subproblems and a one-dimensional subproblem. Simulation results indicate that the proposed algorithm outperforms the state of the arts.

Application of sequential quadratic programming based on active set method in cleaner production

On the platform of general chemical process simulation software (it was named Optimization Engineer, OPEN), a general optimization algorithm for chemical process simulation is developed using C + + code. The algorithm is based on sequential quadratic programming (SQP). We adopt the activity set algorithm and the rotation axis algorithm to generate the activity set to solve the quadratic programming sub-problem. The active set method can simplify the number of constraints and speed up the calculation. At the same time, we used limited memory BFGS algorithm (L-BFGS) to simplify the solution of second derivative matrix. The special matrix storage mode of L-BFGS algorithm can save the storage space and speed up the computing efficiency. We use exact penalty function and traditional step-size rule in the algorithm. These two methods can ensure the convergence of the algorithm, a more correct search direction and suitable search step. The example shows that the advanced optimization function can meet the requirements of General Chemical Process Calculation. The number of iterations can reduce by about 6.0%. The computation time can reduce by about 6.5%. We combined this algorithm with chemical simulation technology to develop the optimization function of chemical engineering simulation. This optimization function can play an important role in the process optimization calculation aiming at energy saving and green production.

A Barzilai and Borwein regularization feasible direction algorithm for convex nonlinear SOC programming with linear constraints

A Barzilai and Borwein regularization feasible direction algorithm is proposed for convex nonlinear second-order cone programming with linear constraints. In the algorithm, a regularization penalty term based on the Barzilai and Borwein parameters is added to the Frank–Wolfe linearization objective function of the direction generating subproblem. The Barzilai and Borwein subproblem is transformed into a multi-block separable convex quadratic second-order cone programming (CQSOCP) subproblem. A parallel inexact alternating direction method is applied to solve the multi-block separable CQSOCP subproblem. The global convergence is given. Numerical results demonstrate that our method is efficient for some random convex nonlinear second-order cone programming problems with low accuracy.

Capacitors placement in distribution systems with nonlinear load by using the variables’ inclusion and interchange algorithm

This work presents a substantial improvement of the variables’ inclusion and interchange algorithm (VIIA) for capacitors placement that considers circuits with harmonic distortion. Several load states are considered, and fixed and switched capacitors are employed in optimization. All the pertinent constraints of voltage magnitude, total harmonic distortion, individual harmonic distortion, and of overstress of capacitors are implemented. The here defined global harmonic-distortion index states the distance to the feasibility or the unfeasibility of a solution with respect the harmonic distortion constraints. The inclusion in the sequential quadratic programming sub-problem of an inequality linear constraint on this global harmonic-distortion index, allows the determining of solutions that comply with the harmonic distortion related constraints. A comparison of the solutions of various examples obtained by the presented method with the best solutions obtained by the Matlab’s genetic algorithm shows the effectiveness of this method.

Continuation methods with the trusty time-stepping scheme for linearly constrained optimization with noisy data

The nonlinear optimization problem with linear constraints has many applications in engineering fields such as the visual-inertial navigation and localization of an unmanned aerial vehicle maintaining the horizontal flight. In order to solve this practical problem efficiently, this paper constructs a continuation method with the trusty time-stepping scheme for the linearly equality-constrained optimization problem at every sampling time. At every iteration, the new method only solves a system of linear equations other than the traditional optimization method such as the sequential quadratic programming (SQP) method, which needs to solve a quadratic programming subproblem. Consequently, the new method can save much more computational time than SQP. Numerical results show that the new method works well for this problem and its consumed time is about one fifth of that of SQP (the built-in subroutine fmincon.m of the MATLAB2018a environment) or that of the traditional dynamical method (the built-in subroutine ode15s.m of the MATLAB2018a environment). Furthermore, we also give the global convergence analysis of the new method.

Novel Relax-and-Retract Algorithm for Intelligent Reflecting Surface Design

In this paper, a novel relax-and-retract algorithm is proposed to tackle the nonconvex unit-modulus constraint in the joint intelligent reflecting surface (IRS) and multiuser multiple-input multiple-output (MU-MIMO) transceiver design. The conventional method to tackle the unit-modulus constraint is semidefinite relaxation (SDR), and its computational complexity is large. By using the symbol detection mean square error (MSE) as the quality of service (QoS), the proposed relax-and-retract approach enables us to get convex quadratically constrained quadratic programming (QCQP) subproblems, which have a much lower computational complexity than the conventional SDR approach. Simulation results show that the proposed relax-and-retract approach has excellent performance in terms of computational complexity, while the transmit power and the unit-modulus hardware implementation are the same as those of the SDR approach.

A QCQP-based splitting SQP algorithm for two-block nonconvex constrained optimization problems with application

This paper discusses a class of two-block nonconvex smooth optimization problems with nonlinear constraints. Based on a quadratically constrained quadratic programming (QCQP) approximation, an augmented Lagrangian function (ALF), and a Lagrangian splitting technique into small-scale subproblems, we propose a novel sequential quadratic programming (SQP) algorithm. First, inspired by the augmented Lagrangian method (ALM), we penalize the quadratic equality constraints associated with the QCQP approximation subproblem in its objective by means of the ALF, and then split the resulting subproblem into two small-scale ones, but both of them are not quadratic programming (QP) due to the square of the quadratic equality constraints in the objective. Second, by ignoring the three-order infinitesimal arising from the squared term, the two small-scale subproblems are reduced to two standard QP subproblems, which can yield an improved search direction. Third, taking the ALF of the discussed problem as a merit function, the next iterate point is generated by the Armijo line search. As a result, a new SQP method, called QCQP-based splitting SQP method, is proposed. Under suitable conditions, the global convergence, strong convergence, iteration complexity and convergence rate of the proposed method are analyzed and obtained. Finally, preliminary numerical experiments and applications were carried out, and these show that the proposed method is promising.

A Globally Convergent SQCQP Method for Multiobjective Optimization Problems

In this article, the concept of the single-objective sequential quadratically constrained quadratic programming method is extended to the multiobjective case and a new line search technique is developed for nonlinear multiobjective optimization problems. The proposed method ensures global convergence as well as spreading of the Pareto front. A descent direction is obtained by solving a quadratically constrained quadratic programming subproblem. A nondifferentiable penalty function is used to restrict the constraint violations. Convergence of the descent sequence is established under the Mangasarian--Fromovitz constraint qualification and some mild assumptions. In addition to this, a new technique is designed for selecting initial points to ensure the spreading of the Pareto front. The method is compared with existing methods using a set of test problems.

A Power Distribution Strategy for Heavy Duty HEV with Series Hybrid Powertrain based on Model Predictive Control Method

Abstract Aiming at the problem of power distribution strategy of heavy-duty HEV with series hybrid powertrain, this paper proposed a power distribution strategy based on model predictive control (MPC). The strategy establishes the related energy consumption index and the physical constraints of the whole vehicle. It predicts the power and speed of vehicle powertrain in a given prediction time domain and distributes generator power and motor power reasonably according to the demand power. In order to reduce the computation of solving quadratic programming subproblems, the improved sequential quadratic programming (ISQP) algorithm is involved to solve the rolling horizon optimization problem. To prove the effectiveness of the strategy, the simulation is carried out under a given condition. The simulation results show that the strategy has better fuel economy than the rule-based strategy under the given condition.

Parameter Optimization and Experimental Analysis of Passive Energy Storage Power-Assisted Exoskeleton

To address the occurrence of lumbar spine disease among labor workers who carry heavy objects, a passive energy storage based exoskeletal apparatus was designed to assist, using springs as energy storage elements and utilizing the change in energy that occurs when the human body is bent during the process of lifting objects. First, the mechanism of the exoskeleton was statically modeled; the spring stiffnesses and the locations of support points were used as design variables to optimize the model by optimizing the effective moment on the lumbar spine. Next, an optimized algorithm (Optdes-Sqp) based on the Newton method for solving quadratic programming subproblems was applied to optimize the stiffnesses of compression and extension springs and the positions of the upper support points of each spring. The accuracy of the simulated model was also verified using MATLAB software. Finally, the effect of optimization was verified, and the respiratory rates and heart rates of subjects before and after wearing the exoskeleton were analyzed. The experimental results show that the exoskeleton designed in this study assisted the subjects, and the results lay the foundation for follow-up designs and studies of exoskeletons.