Showing papers on "Nonlinear programming published in 2015"

Keyframe-based visual-inertial odometry using nonlinear optimization

Abstract: Combining visual and inertial measurements has become popular in mobile robotics, since the two sensing modalities offer complementary characteristics that make them the ideal choice for accurate visual-inertial odometry or simultaneous localization and mapping SLAM. While historically the problem has been addressed with filtering, advancements in visual estimation suggest that nonlinear optimization offers superior accuracy, while still tractable in complexity thanks to the sparsity of the underlying problem. Taking inspiration from these findings, we formulate a rigorously probabilistic cost function that combines reprojection errors of landmarks and inertial terms. The problem is kept tractable and thus ensuring real-time operation by limiting the optimization to a bounded window of keyframes through marginalization. Keyframes may be spaced in time by arbitrary intervals, while still related by linearized inertial terms. We present evaluation results on complementary datasets recorded with our custom-built stereo visual-inertial hardware that accurately synchronizes accelerometer and gyroscope measurements with imagery. A comparison of both a stereo and monocular version of our algorithm with and without online extrinsics estimation is shown with respect to ground truth. Furthermore, we compare the performance to an implementation of a state-of-the-art stochastic cloning sliding-window filter. This competitive reference implementation performs tightly coupled filtering-based visual-inertial odometry. While our approach declaredly demands more computation, we show its superior performance in terms of accuracy.

On-Manifold Preintegration for Real-Time Visual-Inertial Odometry

Abstract: Current approaches for visual-inertial odometry (VIO) are able to attain highly accurate state estimation via nonlinear optimization. However, real-time optimization quickly becomes infeasible as the trajectory grows over time, this problem is further emphasized by the fact that inertial measurements come at high rate, hence leading to fast growth of the number of variables in the optimization. In this paper, we address this issue by preintegrating inertial measurements between selected keyframes into single relative motion constraints. Our first contribution is a \emph{preintegration theory} that properly addresses the manifold structure of the rotation group. We formally discuss the generative measurement model as well as the nature of the rotation noise and derive the expression for the \emph{maximum a posteriori} state estimator. Our theoretical development enables the computation of all necessary Jacobians for the optimization and a-posteriori bias correction in analytic form. The second contribution is to show that the preintegrated IMU model can be seamlessly integrated into a visual-inertial pipeline under the unifying framework of factor graphs. This enables the application of incremental-smoothing algorithms and the use of a \emph{structureless} model for visual measurements, which avoids optimizing over the 3D points, further accelerating the computation. We perform an extensive evaluation of our monocular \VIO pipeline on real and simulated datasets. The results confirm that our modelling effort leads to accurate state estimation in real-time, outperforming state-of-the-art approaches.

Computing in Operations Research using Julia

Abstract: The state of numerical computing is currently characterized by a divide between highly efficient yet typically cumbersome low-level languages such as C, C++, and Fortran and highly expressive yet typically slow high-level languages such as Python and MATLAB. This paper explores how Julia, a modern programming language for numerical computing that claims to bridge this divide by incorporating recent advances in language and compiler design (such as just-in-time compilation), can be used for implementing software and algorithms fundamental to the field of operations research, with a focus on mathematical optimization. In particular, we demonstrate algebraic modeling for linear and nonlinear optimization and a partial implementation of a practical simplex code. Extensive cross-language benchmarks suggest that Julia is capable of obtaining state-of-the-art performance. Data, as supplemental material, are available at http://dx.doi.org/10.1287/ijoc.2014.0623.

A Second-Order Multi-Agent Network for Bound-Constrained Distributed Optimization

Abstract: This technical note presents a second-order multi-agent network for distributed optimization with a sum of convex objective functions subject to bound constraints. In the multi-agent network, the agents connect each others locally as an undirected graph and know only their own objectives and constraints. The multi-agent network is proved to be able to reach consensus to the optimal solution under mild assumptions. Moreover, the consensus of the multi-agent network is converted to the convergence of a dynamical system, which is proved using the Lyapunov method. Compared with existing multi-agent networks for optimization, the second-order multi-agent network herein is capable of solving more general constrained distributed optimization problems. Simulation results on two numerical examples are presented to substantiate the performance and characteristics of the multi-agent network.

Recent advances in trust region algorithms

Abstract: Trust region methods are a class of numerical methods for optimization. Unlike line search type methods where a line search is carried out in each iteration, trust region methods compute a trial step by solving a trust region subproblem where a model function is minimized within a trust region. Due to the trust region constraint, nonconvex models can be used in trust region subproblems, and trust region algorithms can be applied to nonconvex and ill-conditioned problems. Normally it is easier to establish the global convergence of a trust region algorithm than that of its line search counterpart. In the paper, we review recent results on trust region methods for unconstrained optimization, constrained optimization, nonlinear equations and nonlinear least squares, nonsmooth optimization and optimization without derivatives. Results on trust region subproblems and regularization methods are also discussed.

Global supervised descent method

Abstract: Mathematical optimization plays a fundamental role in solving many problems in computer vision (e.g., camera calibration, image alignment, structure from motion). It is generally accepted that second order descent methods are the most robust, fast, and reliable approaches for nonlinear optimization of a general smooth function. However, in the context of computer vision, second order descent methods have two main drawbacks: 1) the function might not be analytically differentiable and numerical approximations are impractical, and 2) the Hessian may be large and not positive definite. Recently, Supervised Descent Method (SDM), a method that learns the “weighted averaged gradients” in a supervised manner has been proposed to solve these issues. However, SDM is a local algorithm and it is likely to average conflicting gradient directions. This paper proposes Global SDM (GSDM), an extension of SDM that divides the search space into regions of similar gradient directions. GSDM provides a better and more efficient strategy to minimize non-linear least squares functions in computer vision problems. We illustrate the effectiveness of GSDM in two problems: non-rigid image alignment and extrinsic camera calibration.

Efficient mixed-integer planning for UAVs in cluttered environments

Abstract: We present a new approach to the design of smooth trajectories for quadrotor unmanned aerial vehicles (UAVs), which are free of collisions with obstacles along their entire length. To avoid the non-convex constraints normally required for obstacle-avoidance, we perform a mixed-integer optimization in which polynomial trajectories are assigned to convex regions which are known to be obstacle-free. Prior approaches have used the faces of the obstacles themselves to define these convex regions. We instead use IRIS, a recently developed technique for greedy convex segmentation [1], to pre-compute convex regions of safe space. This results in a substantially reduced number of integer variables, which improves the speed with which the optimization can be solved to its global optimum, even for tens or hundreds of obstacle faces. In addition, prior approaches have typically enforced obstacle avoidance at a finite set of sample or knot points. We introduce a technique based on sums-of-squares (SOS) programming that allows us to ensure that the entire piecewise polynomial trajectory is free of collisions using convex constraints. We demonstrate this technique in 2D and in 3D using a dynamical model in the Drake toolbox for Matlab [2].

Adaptive firefly algorithm with chaos for mechanical design optimization problems

Abstract: Proposing an extension of firefly algorithmEmployment of picewise chaos, for an further enhanced diversityMaking use of a simple but effective constraint handling methodMaking use of an improved local search procedure Firefly algorithm (FA) is a newer member of bio-inspired meta-heuristics, which was originally proposed to find solutions to continuous optimization problems Popularity of FA has increased recently due to its effectiveness in handling various optimization problems To enhance the performance of the FA even further, an adaptive FA is proposed in this paper to solve mechanical design optimization problems, and the adaptivity is focused on the search mechanism and adaptive parameter settings Moreover, chaotic maps are also embedded into AFA for performance improvement It is shown through experimental tests that some of the best known results are improved by the proposed algorithm

Adjustable Robust Real-Time Power Dispatch With Large-Scale Wind Power Integration

Abstract: In this paper, we propose a robust real-time wind power dispatch framework for coordinating wind farms, automatic generation control (AGC) units, and nonAGC units, which enables wind farms to operate flexibly using maximum power-point tracking strategies. Robust real-time dispatch is formulated as an adjustable robust optimization model incorporating an affinely adjustable controlling strategy compatible with AGC systems. The proposed model can be equivalently transformed to a nonlinear programming problem with linear constraints via duality. The proposed model can also be approximately reduced to a quadratic programming with the objective function simplified. Monte Carlo simulations are carried out to compare the performance of the proposed method against the conventional real-time dispatch scheme. The results show the proposed scheme is robust and reliable.

Convex Optimization of Power Systems

Abstract: Optimization is ubiquitous in power system engineering. Drawing on powerful, modern tools from convex optimization, this rigorous exposition introduces essential techniques for formulating linear, second-order cone, and semidefinite programming approximations to the canonical optimal power flow problem, which lies at the heart of many different power system optimizations. Convex models in each optimization class are then developed in parallel for a variety of practical applications like unit commitment, generation and transmission planning, and nodal pricing. Presenting classical approximations and modern convex relaxations side-by-side, and a selection of problems and worked examples, this is an invaluable resource for students and researchers from industry and academia in power systems, optimization, and control.

Passenger-demands-oriented train scheduling for an urban rail transit network

Abstract: This paper considers the train scheduling problem for an urban rail transit network. We propose an event-driven model that involves three types of events, i.e., departure events, arrival events, and passenger arrival rates change events. The routing of the arriving passengers at transfer stations is also included in the train scheduling model. Moreover, the passenger transfer behavior (i.e., walking times and transfer times of passengers) is also taken into account in the model formulation. The resulting optimization problem is a real-valued nonlinear nonconvex problem. Nonlinear programming approaches (e.g., sequential quadratic programming) and evolutionary algorithms (e.g., genetic algorithms) can be used to solve this train scheduling problem. The effectiveness of the event-driven model is evaluated through a case study.

Deriving robust counterparts of nonlinear uncertain inequalities

Abstract: In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

Decoupled multiagent path planning via incremental sequential convex programming

Abstract: This paper presents a multiagent path planning algorithm based on sequential convex programming (SCP) that finds locally optimal trajectories. Previous work using SCP efficiently computes motion plans in convex spaces with no static obstacles. In many scenarios where the spaces are non-convex, previous SCP-based algorithms failed to find feasible solutions because the convex approximation of collision constraints leads to forming a sequence of infeasible optimization problems. This paper addresses this problem by tightening collision constraints incrementally, thus forming a sequence of more relaxed, feasible intermediate optimization problems.We show that the proposed algorithm increases the probability of finding feasible trajectories by 33% for teams of more than three vehicles in non-convex environments. Further, we show that decoupling the multiagent optimization problem to a number of single-agent optimization problems leads to significant improvement in computational tractability. We develop a decoupled implementation of the proposed algorithm, abbreviated dec-iSCP. We show that dec-iSCP runs 14% faster and finds feasible trajectories with higher probability than a decoupled implementation of previous SCP-based algorithms. The proposed algorithm is real-time implementable and is validated through hardware experiments on a team of quadrotors.

User's Guide for SNOPT Version 7.4: Software for Large-Scale Nonlinear Programming

Abstract: SNOPT is a general-purpose system for constrained optimization. It minimizes a linear or nonlinear function subject to bounds on the variables and sparse linear or nonlinear constraints. It is suitable for large-scale linear and quadratic programming and for linearly constrained optimization, as well as for general nonlinear programs. SNOPT nds solutions that are locally optimal, and ideally any nonlinear functions should be smooth and users should provide gradients. It is often more widely useful. For example, local optima are often global solutions, and discontinuities in the function gradients can often be tolerated if they are not too close to an optimum. Unknown gradients are estimated by nite dierences. SNOPT uses a sequential quadratic programming (SQP) algorithm. Search directions are obtained from QP subproblems that minimize a quadratic model of the Lagrangian function subject to linearized constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point.

Stochastic-Predictive Energy Management System for Isolated Microgrids

Abstract: This paper presents the mathematical formulation and control architecture of a stochastic-predictive energy management system for isolated microgrids. The proposed strategy addresses uncertainty using a two-stage decision process combined with a receding horizon approach. The first stage decision variables (unit commitment) are determined using a stochastic mixed-integer linear programming formulation, whereas the second stage variables (optimal power flow) are refined using a nonlinear programming formulation. This novel approach was tested on a modified CIGRE test system under different configurations comparing the results with respect to a deterministic approach. The results show the appropriateness of the method to account for uncertainty in the power forecast.

Discrete-Time Zhang Neural Network for Online Time-Varying Nonlinear Optimization With Application to Manipulator Motion Generation

Abstract: In this brief, a discrete-time Zhang neural network (DTZNN) model is first proposed, developed, and investigated for online time-varying nonlinear optimization (OTVNO). Then, Newton iteration is shown to be derived from the proposed DTZNN model. In addition, to eliminate the explicit matrix-inversion operation, the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is introduced, which can effectively approximate the inverse of Hessian matrix. A DTZNN-BFGS model is thus proposed and investigated for OTVNO, which is the combination of the DTZNN model and the quasi-Newton BFGS method. In addition, theoretical analyses show that, with step-size $h=1$ and/or with zero initial error, the maximal residual error of the DTZNN model has an $O(\tau ^{2})$ pattern, whereas the maximal residual error of the Newton iteration has an $O(\tau )$ pattern, with $\tau $ denoting the sampling gap. Besides, when $h eq 1$ and $h\in (0,2)$ , the maximal steady-state residual error of the DTZNN model has an $O(\tau ^{2})$ pattern. Finally, an illustrative numerical experiment and an application example to manipulator motion generation are provided and analyzed to substantiate the efficacy of the proposed DTZNN and DTZNN-BFGS models for OTVNO.

A Mixed-Integer Linear Programming Model for the Electric Vehicle Charging Coordination Problem in Unbalanced Electrical Distribution Systems

Abstract: This paper presents a novel mixed-integer linear programming (MILP) model for the electric vehicle charging coordination (EVCC) problem in unbalanced electrical distribution systems (EDSs). Linearization techniques are applied over a mixed-integer nonlinear programming model to obtain the proposed MILP formulation based on current injections. The expressions used to represent the steady-state operation of the EDS take into account a three-phase representation of the circuits, as well as the imbalance of the loads, leading to a more realistic model. Additionally, the proposed formulation considers the presence of distributed generators and operational constraints such as voltage and current magnitude limits. The optimal solution for the mathematical model was found using commercial MILP solvers. The proposed formulation was tested in a distribution system used in the specialized literature. The results show the efficiency and the robustness of the methodology, and also demonstrate that the model can be used in the solution of the EVCC problem in EDSs.

Eco-Departure of Connected Vehicles With V2X Communication at Signalized Intersections

Abstract: Eco-driving at signalized intersections has significant potential for energy saving. In this paper, we focus on eco-departure operations of connected vehicles equipped with an internal combustion engine and a step-gear automatic transmission. A Bolza-type optimal control problem (OCP) is formulated to minimize engine fuel consumption. Due to the discrete gear ratio, this OCP is a nonlinear mixed-integer problem, which is challenging to handle by most existing optimization methods. The Legendre pseudospectral method combining the knotting technique is employed to convert it into a multistage interconnected nonlinear programming problem, which then solves the optimal engine torque and transmission gear position. The fuel-saving benefit of the optimized eco-departing operation is validated by a passenger car with a five-speed transmission. For real-time implementation, a near-optimal departing strategy is proposed to quickly determine the behavior of the engine and transmission. When a string of vehicles are departing from an intersection, the acceleration of the leading vehicle(s) should be considered to control the following vehicles. This issue is also addressed in this paper.

Validation of nominations in gas network optimization: models, methods, and solutions

Abstract: In this article, we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult nonconvex mixedinteger nonlinear feasibility problem. The first phase consists of four distinct algorithms applying mixedinteger linear, mixedinteger nonlinear, nonlinear, and methods for complementarity constraints to compute possible settings for the discrete decisions. The second phase employs a precise continuous nonlinear programming model of the gas network. Using this setup, we are able to compute high-quality solutions to real-world industrial instances that are si...

Global optimization method for network design problem with stochastic user equilibrium

Abstract: In this paper, we consider the continuous road network design problem with stochastic user equilibrium constraint that aims to optimize the network performance via road capacity expansion. The network flow pattern is subject to stochastic user equilibrium, specifically, the logit route choice model. The resulting formulation, a nonlinear nonconvex programming problem, is firstly transformed into a nonlinear program with only logarithmic functions as nonlinear terms, for which a tight linear programming relaxation is derived by using an outer-approximation technique. The linear programming relaxation is then embedded within a global optimization solution algorithm based on range reduction technique, and the proposed approach is proved to converge to a global optimum.

Development and Application of the Bat Algorithm for Optimizing the Operation of Reservoir Systems

Abstract: Optimal utilization of water resources by means of water transfers and reservoirs in semiarid and arid regions is used to mitigate natural water scarcity. In this context, metaheuristic algorithms for optimum reservoir system operation have become an attractive alternative to traditional operations research algorithms such as linear programming (LP), nonlinear programming (NLP), and dynamic programming (DP). This paper presents the metaheuristic bat algorithm (BA) and its application to the optimal operation of the Karoun-4 reservoir system in Iran and to a hypothetical four-reservoir system. The merits of the performance of the BA in the optimization of reservoir operation are demonstrated by comparison to those of LP, NLP, and genetic algorithm (GA) in terms of the convergence to global optima and of the variance of results about global optima for reservoir optimization problems.

A Two-Layer Recurrent Neural Network for Nonsmooth Convex Optimization Problems

Abstract: In this paper, a two-layer recurrent neural network is proposed to solve the nonsmooth convex optimization problem subject to convex inequality and linear equality constraints. Compared with existing neural network models, the proposed neural network has a low model complexity and avoids penalty parameters. It is proved that from any initial point, the state of the proposed neural network reaches the equality feasible region in finite time and stays there thereafter. Moreover, the state is unique if the initial point lies in the equality feasible region. The equilibrium point set of the proposed neural network is proved to be equivalent to the Karush–Kuhn–Tucker optimality set of the original optimization problem. It is further proved that the equilibrium point of the proposed neural network is stable in the sense of Lyapunov. Moreover, from any initial point, the state is proved to be convergent to an equilibrium point of the proposed neural network. Finally, as applications, the proposed neural network is used to solve nonlinear convex programming with linear constraints and $L_{1}$ -norm minimization problems.

Benchmarking optimization solvers for structural topology optimization

Abstract: The purpose of this article is to benchmark different optimization solvers when applied to various finite element based structural topology optimization problems. An extensive and representative library of minimum compliance, minimum volume, and mechanism design problem instances for different sizes is developed for this benchmarking. The problems are based on a material interpolation scheme combined with a density filter. Different optimization solvers including Optimality Criteria (OC), the Method of Moving Asymptotes (MMA) and its globally convergent version GCMMA, the interior point solvers in IPOPT and FMINCON, and the sequential quadratic programming method in SNOPT, are benchmarked on the library using performance profiles. Whenever possible the methods are applied to both the nested and the Simultaneous Analysis and Design (SAND) formulations of the problem. The performance profiles conclude that general solvers are as efficient and reliable as classical structural topology optimization solvers. Moreover, the use of the exact Hessians in SAND formulations, generally produce designs with better objective function values. However, with the benchmarked implementations solving SAND formulations consumes more computational time than solving the corresponding nested formulations.

Power Distribution Network Expansion Planning Considering Distribution Automation

Abstract: This paper presents a novel profit-based model for multistage distribution network expansion planning. In the proposed model, some capabilities of the distribution automation system are considered to achieve a plan that is more compatible with the strategic plan towards a smart distribution grid. Moreover, a method for reliability evaluation of a distribution network in the planning studies is proposed. The objective function of the planning problem is the net present value of the company's profit. The problem is solved by employing the genetic algorithm approach as a promising technique in solving the mixed-integer nonlinear programming problem. The proposed method is evaluated on an illustrative test network and the obtained results are presented and discussed.