Showing papers on "Quadratically constrained quadratic program published in 2019"

From the Racetrack to the Road: Real-Time Trajectory Replanning for Autonomous Driving

Abstract: In emergency situations, autonomous vehicles will be forced to operate at their friction limits in order to avoid collisions. In these scenarios, coordinating the planning of the vehicle's path and speed gives the vehicle the best chance of avoiding an obstacle. Fast reaction time is also important in an emergency, but approaches to the trajectory planning problem based on nonlinear optimization are computationally expensive. This paper presents a new scheme that simultaneously modifies the desired path and speed profile for a vehicle in response to the appearance of an obstacle, significant tracking error, or other environmental change. By formulating the trajectory optimization problem as a quadratically constrained quadratic program, solution times of less than 20 ms are possible even with a 10-s planning horizon. A simplified point mass model is used to describe the vehicle's motion, but the incorporation of longitudinal weight transfer and road topography means that the vehicle's acceleration limits are modeled more accurately than in comparable approaches. Experimental data from an autonomous vehicle in two scenarios demonstrate how the trajectory planner enables the vehicle to avoid an obstacle even when the obstacle appears suddenly and the vehicle is already operating near the friction limits.

Certifiably Globally Optimal Extrinsic Calibration From Per-Sensor Egomotion

Abstract: We present a certifiably globally optimal algorithm for determining the extrinsic calibration between two sensors that are capable of producing independent egomotion estimates. This problem has been previously solved using a variety of techniques, including local optimization approaches that have no formal global optimality guarantees. We use a quadratic objective function to formulate calibration as a quadratically constrained quadratic program (QCQP). By leveraging recent advances in the optimization of QCQPs, we are able to use existing semidefinite program solvers to obtain a certifiably global optimum via the Lagrangian dual problem. Our problem formulation can be globally optimized by existing general-purpose solvers in less than a second, regardless of the number of measurements available and the noise level. This enables a variety of robotic platforms to rapidly and robustly compute and certify a globally optimal set of calibration parameters without a prior estimate or operator intervention. We compare the performance of our approach with a local solver on extensive simulations and multiple real datasets. Finally, we present necessary observability conditions that connect our approach to recent theoretical results and analytically support the empirical performance of our system.

Global solution of non-convex quadratically constrained quadratic programs

Abstract: The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P∗). This equivalent problem (P∗) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y=xxT, where x are the initial variables of problem (P). We then propose to solve (P∗) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatia...

Bilevel programming approach to demand response management with day-ahead tariff

Abstract: This paper introduces a bilevel programming approach to electricity tariff optimization for the purpose of demand response management (DRM) in smart grids. In the multi-follower Stackelberg game model, the leader is the profit-maximizing electricity retailer, who must set a time-of-use variable energy tariff in the grid. Followers correspond to groups of prosumers (simultaneous producers and consumers of the electricity. They response to the observed tariff, schedule controllable loads and determine the charging/discharging policy of their batteries to minimize the cost of electricity and to maximize the utility at the same time. A bilevel programming formulation of the problem is defined, and its fundamental properties are proven. The primal-dual reformulation is proposed in this paper to convert the bilevel optimization problem into a single-level quadratically constrained quadratic program (QCQP), and a successive linear programming (SLP) algorithm is applied to solve it. It is demonstrated in computational experiments that the proposed approach outperforms typical earlier methods based on the Karush–Kuhn–Tucker (KKT) reformulation regarding both solution quality and computational efficiency on practically relevant problem sizes. Besides, it also offers more flexible modeling capabilities.

Rank-One Solutions for SDP Relaxation of QCQPs in Power Systems

Abstract: It has been shown that a large number of computationally difficult problems can be equivalently reformulated into quadratically constrained quadratic programs (QCQPs) in the literature of power systems. Due to the NP-hardness of general QCQPs, main effort of this stream of problems has been put into deriving near-optimal solutions with low computational complexity. Recently, semidefinite programming (SDP) relaxation has been recognized as a promising technique to solve QCQPs from various applications such as the alternating current (ac) optimal power flow (OPF) problem. However, this technique has not been guaranteed to achieve a rank-one solution, which is a necessary condition to recover a feasible solution of the original QCQPs. In this paper, instead of investigating the conditions under which a rank-one solution exists, we propose a general solution framework to derive near-optimal but rank-one solutions for the SDP relaxation of QCQPs. In the proposed algorithm, all the parameters are provided in a systematic manner. In order to demonstrate the effectiveness of our method, the proposed algorithm is applied to solve the ac-OPF and state estimation problems in various settings. Extensive numerical results show that our method succeeds in obtaining rank-one solutions in all our case studies and only small optimality gaps are induced by our approach.

Data-Driven Reserve Allocation With Frequency Security Constraint Considering Inverter Air Conditioners

Abstract: Inverter air conditioners (IACs) with considerable total capacity and fast response speed are ideal demand response resources, which are of significant potential to provide reserve capacity for the power system frequency regulation. However, due to the complexity and implicitness of the frequency response models, it is difficult to formulate the optimization problem considering frequency dynamics to allocate reserve capacity precisely. In this paper, a data-driven method is proposed for reserve allocation with the frequency security constraint considering IACs. Firstly, the equivalent frequency response model of aggregated IACs is developed considering electrical-thermal characteristics and then incorporated into the frequency regulation framework of power systems along with conventional generators. Then, simulations are implemented to generate massive reserve samples with deterministic frequency security labels. Later, a support vector machine (SVM) based frequency security classifier is trained to convert the implicit frequency security constraint into polynomials and reshape the reserve allocation problem into a solvable general quadratically constrained quadratic program (QCQP). Finally, a heuristic Suggest-and-Improve (SI) method is adopted to deal with the nonconvex QCQP of interest. It is demonstrated by numerical studies that the proposed data-driven method enables power systems to operate closer to the frequency security boundaries and thus achieve lower costs.

Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming

Abstract: In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations of QCQP using the reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP, except for direct lifting and linearization. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some new effective valid constraints. Moreover, we extend the reach of the RLT-like techniques for almost all different types of constraint-pairs (including valid inequalities by linearizing the product of a pair of SOC constraints, and the Hadamard product or the Kronecker product of two respective valid linear matrix inequalities), examine dominance relationships among different valid inequalities, and explore almost all possibilities of gaining benefits from generating valid constraints. We also successfully demonstrate that applying RLT-like techniques to additional redundant linear constraints could reduce the relaxation gap significantly. We demonstrate the efficiency of our results with numerical experiments.

Nonlinear Model Predictive Control for Robust Bipedal Locomotion Exploring CoM Height and Angular Momentum Changes.

Abstract: Human beings can make use of various reactive strategies, e.g. foot location adjustment and upper-body inclination, to keep balance while walking under dynamic disturbances. In this work, we propose a novel Nonlinear Model Predictive Control (NMPC) framework for versatile bipedal gait pattern generation, with the capabilities of footstep adjustment, Center of Mass (CoM) height variation and angular momentum adaptation. These features are realized by constraining the Zero Moment Point motion with considering the variable CoM height and angular momentum change of the Inverted Pendulum plus Flywheel Model. In addition, the NMPC framework also takes into account the constraints of footstep location, CoM vertical motion, upper-body inclination and joint torques, and is finally formulated as a quadratically constrained quadratic program. Therefore, it can be solved efficiently by Sequential Quadratic Programming. Using this unified framework, versatile walking pattern with exploiting time-varying CoM height trajectory and angular momentum changes can be generated based only on the terrain information input. Furthermore, the improved capability for balance recovery under external pushes has been demonstrated through simulation studies.

A quadratic programming model for the optimization of off-gas networks in integrated steelworks

Abstract: The European steel industry is constantly promoting developments, which can increase efficiency and lower the environmental impact of the steel production processes. In particular, a strong focus refers to the minimization of the energy consumption. This paper presents part of the work of the research project entitled “Optimization of the management of the process gas network within the integrated steelworks” (GASNET), which aims at developing a decision support system supporting energy managers and other concerned technical personnel in the implementation of an optimized off-gases management and exploitation considering environmental and economic objectives. A mathematical model of the network as a capacitated digraph with costs on arcs is proposed and an optimization problem is formulated. The objective of the optimization consists in minimizing the wastes of process gases and maximizing the incomes. Several production constraints need to be accounted. In particular, different types of gases are mixing in the same network. The constraints that model the mixing make the problem computationally difficult: it is a non-convex quadratically constrained quadratic program (QCQP). Two formulations of the problem are presented: the first one is a minimum cost flow problem, which is a linear program and is thus computationally fast to solve, but suitable only for a single gas network. The second formulation is a quadratically constrained quadratic program, which is slower, but covers more general cases, such as the ones, which are characterized by the interaction among multiple gas networks. A user-friendly graphical interface has been developed and tests over existing plant networks are performed and analyzed.

A study of rank-one sets with linear side constraints and application to the pooling problem

Abstract: We study sets defined as the intersection of a rank-1 constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-1 set is polyhedral or second-order cone representable. In all these cases, we also show that a linear objective can be optimized in polynomial time over these sets. Towards the application side, we show how these sets relate to commonly occurring substructures of a general quadratically constrained quadratic program. To further illustrate the benefit of studying quadratically constrained quadratic programs from a rank-1 perspective, we propose new rank-1 formulations for the generalized pooling problem and use our convexification results to obtain several new convex relaxations for the pooling problem. Finally, we run a comprehensive set of computational experiments and show that our convexification results together with discretization significantly help in improving dual bounds for the generalized pooling problem.

Energy Efficient Transceiver Design in MIMO Interference Channels: The Selfish, Unselfish, Worst-Case, and Robust Methods

Abstract: This paper is concerned with the energy efficiency maximization problem in multiple-input multiple-output interference channels. In the first part of this paper, we design the transmit beamforming matrices based on three approaches. In the first approach, each user tries to maximize their own energy efficiency, resulting in what we call the selfish method. The maximization problem corresponding to the selfish method can be implemented in a distributed manner with a semi-closed-form solution. The second approach aims to maximize the weighted sum-energy efficiency; we refer to it as the unselfish method. The proposed unselfish method has the following advantages. 1) It is easily implementable as the problem can be divided into several sub-problems and solved in a distributed manner. 2) It results in a quadratically constrained quadratic program with a semi-closed-form solution. Also, we consider fairness with the minimum-energy efficiency maximization problem, an approach that results in what we call the worst-case method. In the second part, general energy efficiency maximization in the presence of norm-bounded channel uncertainty is investigated which leads to a semi-definite program; the resulting method is called the robust method. Finally, the effect of the system parameters is evaluated through simulations.

Novel Approach Towards Global Optimality of Optimal Power Flow Using Quadratic Convex Optimization

Abstract: Optimal Power Flow (OPF) can be modeled as a nonconvex Quadratically Constrained Quadratic Program (QCQP). Our purpose is to solve OPF to global optimality. To this end, we specialize the Mixed-Integer Quadratic Convex Reformulation method (MIQCR) to (OPF). This is a method in two steps. First, a Semi-Definite Programming (SDP) relaxation of (OPF) is solved. Then the optimal dual variables of this relaxation are used to reformulate OPF into an equivalent new quadratic program, where all the non-convexity is moved to one additional constraint. In the second step, this reformulation is solved within a branch-and-bound algorithm, where at each node a quadratic and convex relaxation of the reformulated problem, obtained by relaxing the non-convex added constraint, is solved. The key point of our approach is that the lower bound at the root node of the branch-and-bound tree is equal to the SDP relaxation value. We test this method on several OPF cases, from two-bus networks to more-than-a-thousand-buses networks from the MAT-POWER repository. Our first results are very encouraging.

Mvdr Robust Adaptive Beamforming Design with Direction of Arrival and Generalized Similarity Constraints

Abstract: The MVDR robust adaptive beamforming design problem based on estimation of the signal-of-interest (SOI) steering vector is considered. In this case, the optimal beamformer is obtained by computing the sample matrix inverse and an optimal estimate of the SOI steering vector. In order to find the optimal steering vector estimate of the SOI, a new beamformer output power maximization problem is formulated subject to a double-sided norm perturbation constraint, a generalized similarity constraint, and a direction-of-arrival (DOA) constraint that guarantees that the DOA of the SOI is away from the DOA region of all linear combinations of the interference steering vectors. It turns out that the power maximization problem is a nonconvex quadratically constrained quadratic program (QCQP) with two homogenous and one inhomogeneous constraints. In general, a globally optimal solution for the QCQP is not guaranteed; however, we herein derive sufficient optimality conditions to ensure the existence of an optimal solution, and develop an efficient algorithm to find the solution. To validate our results, simulation examples are presented, and they demonstrate the improved performance of the new robust adaptive beamformer in terms of the output SINR.

Regression via Kirszbraun Extension with Applications to Imitation Learning.

Abstract: Learning by demonstration is a versatile and rapid mechanism for transferring motor skills from a teacher to a learner. A particular challenge in imitation learning is the so-called correspondence problem, which involves mapping actions between a teacher and a learner having substantially different embodiments (say, human to robot). We present a general, model free and non-parametric imitation learning algorithm based on regression between two Hilbert spaces. We accomplish this via Kirszbraun's extension theorem --- apparently the first application of this technique to supervised learning --- and analyze its statistical and computational aspects. We begin by formulating the correspondence problem in terms of quadratically constrained quadratic program (QCQP) regression. Then we describe a procedure for smoothing the training data, which amounts to regularizing hypothesis complexity via its Lipschitz constant. The Lipschitz constant is tuned via a Structural Risk Minimization (SRM) procedure, based on the covering-number risk bounds we derive. We apply our technique to a static posture imitation task between two robotic manipulators with different embodiments, and report promising results.

A QCQP and SDP Formulation of the Optimal Power Flow Including Renewable Energy Resources

Abstract: The optimal power flow (OPF) problem is a relevant subject for the secure and economic power systems operation. For instance, OPF can be used to reduce the power system technical losses. A reduction in 0.1% in the losses accounts for near 50 billion USD in cost savings. Likewise, OPF can be used along with power production forecast tools, for renewable energy sources, to evaluate their impact in the grid security operation planning. In an OPF problem, an objective function related with demand supply generation cost, power line losses, or violation limits is optimized subject to several system security constraints. These constraints are related with branch power flow limits, voltages limits, power injection limits, among others. The exact model of the objective function and constraints of the OPF problem is neither linear nor convex. In this paper it is presented the OPF problem in a Quadratically Constrained Quadratic Program (QCQP) approach. Objective function is based on a quadratic function of the bus voltages and constraints are formulated as Quadratic forms. A two-bus system is used to demonstrates the non-convexity of the OPF problem. Also, it is presented the Rank- 1 convex relaxation of the OPF problem which relaxes the QCQP model into a positive Semidefinite Programming (SDP) model. Once the OPF problem is relaxed into a SDP convex form, a global optimal solution can be obtained. An application example for the OPF problem is presented for the IEEE 14 system. The QCQP and the SDP results are compared and discussed.

Optimization Design of M-Channel Oversampled Graph Filter Banks via Successive Convex Approximation

Abstract: This letter presents an efficient algorithm to design an M-channel oversampled graph filter bank, whose overall performance is governed by the spectral characteristic of the filters and the reconstruction error. The design of the filters is formulated into a constrained optimization problem that minimizes the stopband energy of the filters subject to the reconstruction error constraint. The problem is a non-convex quadratically constrained quadratic program (QCQP), which is typically NP-hard. In order to overcome this challenge, we apply the successive convex approximation to successively solve it. At each iteration, the non-convex QCQP is approximately transformed into the convex one by linearizing the constraint. Numerical examples and comparison are included to show that the proposed approach can lead to the oversampled graph filter banks with satisfactory overall performance, particularly good spectral selectivity.

Novel Approach Towards Global Optimality of Optimal Power Flow Using Quadratic Convex Optimization

Abstract: Optimal Power Flow (OPF) can be modeled as a non-convex Quadratically Constrained Quadratic Program (QCQP). Our purpose is to solve OPF to global optimality. To this end, we specialize the Mixed-Integer Quadratic Convex Reformulation method (MIQCR) to (OPF). This is a method in two steps. First, a Semi-Definite Programming (SDP) relaxation of (OPF) is solved. Then the optimal dual variables of this relaxation are used to reformulate OPF into an equivalent new quadratic program, where all the non-convexity is moved to one additional constraint. In the second step, this reformulation is solved within a branch-and-bound algorithm, where at each node a quadratic and convex relaxation of the reformulated problem, obtained by relaxing the non-convex added constraint, is solved. The key point of our approach is that the lower bound at the root node of the branch-and-bound tree is equal to the SDP relaxation value. We test this method on several OPF cases, from two-bus networks to more-than-a-thousand-buses networks from the MAT-POWER repository. Our first results are very encouraging.

Energy Efficient Precoder in Multi-User MIMO Systems With Imperfect Channel State Information

Abstract: This letter is on the energy efficient precoder design in multi-user multiple-input multiple-output systems which is also robust with respect to the imperfect channel state information at the transmitters. In other words, we design the precoder matrix associated with each transmitter to maximize the general energy efficiency of the network. We investigate the problem in two conventional cases. The first case considers the statistical characterization for the channel estimation error that leads to a quadratically constrained quadratic program with a semi-closed-form solution. Then, we turn our attention to the case which considers only the uncertainty region for the channel estimation error; this case eventually results in a semi-definite program.

Optimal State Feedback Controller for a Nuclear Reactor

Abstract: This article considers the design of an optimal state feedback controller for a nuclear reactor. The model of 700-MWe Indian pressurized heavy water-type reactor is described and is considered for design. The cost function is the norm of the state feedback vector, which is minimized by making use of extra degrees of freedom obtained due to nonspecific pole placement in the left half of the ${s}$ plane. The stability region is approximated by a maximal sphere in the co-efficient space of the closed-loop characteristic polynomial. The optimization problem is formulated as a quadratically constrained quadratic program and is solved using a primal-dual search method. The issues related to the choice of the center of the approximated sphere and correct step size in the search method are addressed. The norm of the state feedback vector obtained by this method is about 100 times less than those obtained using comparable methods and yet yields acceptable transient performance at various operating points of the reactor.

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

Abstract: 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.

Multiple Kernel Learning-Based Uncertainty Set Construction for Robust optimization

Abstract: In robust optimization (RO), a focal point is the design of uncertainty set that delineates possible realizations of uncertainty since it heavily impacts the robustness of solutions. We propose in this paper a multiple kernel learning (MKL) based support vector clustering (SVC) method for polytopic uncertainty set construction in data-driven RO. By assuming a set of candidate piecewise linear kernel functions, the MKL framework not only derives an enclosing sphere in the input space, but also automatically derives optimal coefficients of kernel functions by only solving a quadratically constrained quadratic program. The learnt sphere turns out to be a compact polyhedral uncertainty set to be used in RO, which helps reducing the conservatism of robust solutions. Meanwhile, although massive data samples and kernel functions are used in MKL, the induced polytopic uncertainty set tends to have a succinct expression, thereby well preserving the tractability of the induced optimization problem. It also allows a decisionmaker to conveniently adjust the conservatism of the data-driven uncertainty set by manipulating only one parameter, which is user-friendly in practice. Numerical case studies are carried out to demonstrate the potential advantages of the proposed method in promoting the practicability of RO techniques.

Robust Feasibility of Systems of Quadratic Equations Using Topological Degree Theory

Abstract: We consider the problem of measuring the margin of robust feasibility of solutions to a system of nonlinear equations. We study the special case of a system of quadratic equations, which shows up in many practical applications such as the power grid and other infrastructure networks. This problem is a generalization of quadratically constrained quadratic programming (QCQP), which is NP-Hard in the general setting. We develop approaches based on topological degree theory to estimate bounds on the robustness margin of such systems. Our methods use tools from convex analysis and optimization theory to cast the problems of checking the conditions for robust feasibility as a nonlinear optimization problem. We then develop inner bound and outer bound procedures for this optimization problem, which could be solved efficiently to derive lower and upper bounds, respectively, for the margin of robust feasibility. We evaluate our approach numerically on standard instances taken from the MATPOWER database of AC power flow equations that describe the steady state of the power grid. The results demonstrate that our approach can produce tight lower and upper bounds on the margin of robust feasibility for such instances.

Semidefinite programming relaxations through quadratic reformulation for box-constrained polynomial optimization problems

Abstract: In this paper we introduce new semidefinite programming relaxations to box-constrained polynomial optimization programs ( $P$ ). For this, we first reformulate ( $P$ ) into a quadratic program. More precisely, we recursively reduce the degree of ( $P$ ) to two by substituting the product of two variables by a new one. We obtain a quadratically constrained quadratic program. We build a first immediate SDP relaxation in the dimension of the total number of variables. We then strengthen the SDP relaxation by use of valid constraints that follow from the quadratization. We finally show the tightness of our relaxations through several experiments on box polynomial instances.

Optimal finite-dimensional spectral densities for the identification of continuous-time MIMO systems

Abstract: This paper presents a method for designing inputs to identify linear continuous-time multiple-input multiple-output (MIMO) systems. The goal here is to design a T-optimal band-limited spectrum satisfying certain input/output power constraints. The input power spectral density matrix is parametrized as the product $$\phi_u(\text{j}\omega)=\frac{1}{2}H(\text{j}\omega)H^\text{H}(\text{j}\omega)$$ , where H(jω) is a matrix polynomial. This parametrization transforms the T-optimal cost function and the constraints into a quadratically constrained quadratic program (QCQP). The QCQP turns out to be a non-convex semidefinite program with a rank one constraint. A convex relaxation of the problem is first solved. A rank one solution is constructed from the solution to the relaxed problem. This relaxation admits no gap between its solution and the original non-convex QCQP problem. The constructed rank one solution leads to a spectrum that is optimal. The proposed input design methodology is experimentally validated on a cantilever beam bonded with piezoelectric plates for sensing and actuation. Subspace identification algorithm is used to estimate the system from the input-output data.

Regression via Kirszbraun Extension

Abstract: We present a framework for performing regression between two Hilbert spaces. We accomplish this via Kirszbraun's extension theorem -- apparently the first application of this technique to supervised learning -- and analyze its statistical and computational aspects. We begin by formulating the correspondence problem in terms of quadratically constrained quadratic program (QCQP) regression. Then we describe a procedure for smoothing the training data, which amounts to regularizing hypothesis complexity via its Lipschitz constant. The Lipschitz constant is tuned via a Structural Risk Minimization (SRM) procedure, based on the covering-number risk bounds we derive. We apply our technique to learn a transformation between two robotic manipulators with different embodiments, and report promising results.

New Computational and Statistical Aspects of Regularized Regression with Application to Rare Feature Selection and Aggregation

Abstract: Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a unified computational framework for defining norms that promote such structures. More specifically, we develop associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models. As an example, we study a norm, which we term the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. We further discuss how the K-means algorithm can serve as the underlying projection oracle in this case and how it can be efficiently represented as a quadratically constrained quadratic program. Our motivation for the study of this norm is regularized regression in the presence of rare features which poses a challenge to various methods within high-dimensional statistics, and in machine learning in general. The proposed estimation procedure is designed to perform automatic feature selection and aggregation for which we develop statistical bounds. The bounds are general and offer a statistical framework for norm-based regularization. The bounds rely on novel geometric quantities on which we attempt to elaborate as well.

Nonlinear Model Predictive Control for Robust Bipedal Locomotion: Exploring Angular Momentum and CoM Height Changes

Abstract: Human beings can utilize multiple balance strategies, e.g. step location adjustment and angular momentum adaptation, to maintain balance when walking under dynamic disturbances. In this work, we propose a novel Nonlinear Model Predictive Control (NMPC) framework for robust locomotion, with the capabilities of step location adjustment, Center of Mass (CoM) height variation, and angular momentum adaptation. These features are realized by constraining the Zero Moment Point within the support polygon. By using the nonlinear inverted pendulum plus flywheel model, the effects of upper-body rotation and vertical height motion are considered. As a result, the NMPC is formulated as a quadratically constrained quadratic program problem, which is solved fast by sequential quadratic programming. Using this unified framework, robust walking patterns that exploit reactive stepping, body inclination, and CoM height variation are generated based on the state estimation. The adaptability for bipedal walking in multiple scenarios has been demonstrated through simulation studies.

A Decomposition Method for Large-scale Convex Quadratically Constrained Quadratic Programs.

Abstract: We consider solving a convex quadratically constrained quadratic program (QCQP), which has a wide range of applications, including machine learning, data analysis and signal processing. While small to mid-sized convex QCQPs can be solved efficiently by interior-point algorithms, large-scale problems pose significant challenges to traditional centralized algorithms, since the exploding volume of data may overwhelm a single computing unit. In this paper, we propose a decomposition method for general non-separable, large-scale convex QCQPs, using the idea of predictor-corrector proximal primal-dual update with an adaptive step size. The algorithm enables distributed storage of data as well as distributed computing. We both establish convergence of the algorithm to a global optimum and test the algorithm on a computer cluster with multiple threads. The numerical test is done on data sets of different scales using Message Passing Interface, and the results show that our algorithm exhibits favourable scalability for large-scale data even when CPLEX fails to provide a solution due to memory limits.

L1-norm fitting of elliptic paraboloids with prior information for enhanced coniferous tree localization in als point clouds

Abstract: . Airborne laser scanning (ALS) is an established tool for deriving various tree characteristics in forests. In some applications, an accurate pointwise estimate of the tree position is required. For dense data acquired by TLS or UAV-mounted scanners, this can be achieved by locating the stem, whose center coordinates are then used for deriving the planimetric tree position. However, in case of standard ALS data this is often not an option due to the low probability of obtaining stem hits in operational scenarios of forest mapping campaigns. This paper presents an alternative, indirect approach where the tree position is approximated as the center of a quadric surface which best represents the tree crown shape. The study targets coniferous trees due to their distinct crown shape which may be approximated by an elliptic paraboloid. It is assumed that individual tree point clusters are given and the task is to find the tree center for each cluster. We first consider the general problem of fitting an elliptic paraboloid with a known axis and an L1 residual norm error criterion, which is more robust to outliers compared to least-squares fitting. We formulate this problem as a quadratically constrained quadratic program (QCQP), and show how prior knowledge on the crown shape and center position can be incorporated. Next, a computationally simpler problem is considered where the paraboloid semiaxis lengths are constrained to be equal, and a corresponding linear program is constructed. Experiments on ALS datasets of forest plots from Bavaria, Germany and Oregon, USA reveal that a reduction in median tree position error of up to 20% can be attained compared to both least-squares fitting and other baseline techniques, resulting in an absolute error of ca. 22 cm on both datasets.