arXiv subjects

Ramtin Madani

Publications and source records attributed to Ramtin Madani.

At least 19 recordsLinked to original sources

arXiv2022

Parabolic Relaxation for Quadratically-constrained Quadratic Programming -- Part I: Definitions & Basic Properties

For general quadratically-constrained quadratic programming (QCQP), we propose a parabolic relaxation described with convex quadratic constraints. An interesting property of the parabolic relaxation is that the original non-convex feasible set is contained on the boundary of the parabolic relaxation. Under certain assumptions, this property enables one to recover near-optimal feasible points via objective penalization. Moreover, through an appropriate change of coordinates that requires a one-time computation of an optimal basis, the easier-to-solve parabolic relaxation can be made as strong as a semidefinite programming (SDP) relaxation, which can be effective in accelerating algorithms that require solving a sequence of convex surrogates. The majority of theoretical and computational results are given in the next part of this work [57].

math.OC↗

arXiv2022

Parabolic Relaxation for Quadratically-constrained Quadratic Programming -- Part II: Theoretical & Computational Results

In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions. In this second part, we show that starting from a feasible solution or a near-feasible solution satisfying certain regularity conditions, the sequential penalized parabolic relaxation algorithm convergences to a point which satisfies Karush-Kuhn-Tucker optimality conditions. Next, we present numerical experiments on benchmark non-convex QCQP problems as well as large-scale instances of system identification problem demonstrating the efficiency of the proposed approach.

math.OC↗

arXiv2022

Proactive Posturing of Large Power Grid for Mitigating Hurricane Impacts

In the past decade, natural disasters such as hurricanes have challenged the operation and control of U.S. power grid. It is crucial to develop proactive strategies to assist grid operators for better emergency response and minimized electricity service interruptions; the better the grid may be preserved, the faster the grid can be restored. In this paper, we propose a proactive posturing of power system elements, and formulate a Security-Constrained Optimal Power Flow (SCOPF) informed by cross-domain hurricane modeling as well as its potential impacts on grid elements. Simulation results based on real-world power grid and historical hurricane event have verified the applicability of the proposed optimization formulation, which shows potential to enable grid operators and planners with interactive cross-domain data analytics for mitigating hurricane impacts.

eess.SY↗

arXiv2022

A First-Order Numerical Algorithm without Matrix Operations

This paper offers a matrix-free first-order numerical method to solve large-scale conic optimization problems. Solving systems of linear equations pose the most computationally challenging part in both first-order and second-order numerical algorithms. Existing direct and indirect methods are either computationally expensive or compromise on solution accuracy. Alternatively, we propose an easy-to-compute decomposition method to solve sparse linear systems that arise in conic optimization problems. Its iterations are tractable, highly parallelizable, with closed-form solutions. This algorithm can be easily implemented on distributed platforms, such as graphics processing units, with orders-of-magnitude time improvement. The performance of the proposed solver is demonstrated on large-scale conic optimization problems and is compared with the state-of-the-art first-order solvers.

math.OC↗

arXiv2022

$Δ$-MILP: Deep Space Network Scheduling via Mixed-Integer Linear Programming

This paper introduces $Δ$-MILP, a powerful variant of the mixed-integer linear programming (MILP) optimization framework to solve NASA's Deep Space Network (DSN) scheduling problem. This work is an extension of our original MILP framework (DOI:10.1109/ACCESS.2021.3064928) and inherits many of its constructions and strengths, including the base MILP formulation for DSN scheduling. To provide more feasible schedules with respect to the DSN requirements, $Δ$-MILP incorporates new sets of constraints including 1) splitting larger tracks into shorter segments and 2) preventing overlapping between tracks on different antennas. Additionally, $Δ$-MILP leverages a heuristic to balance mission satisfaction and allows to prioritize certain missions in special scenarios including emergencies and landings. Numerical validations demonstrate that $Δ$-MILP now satisfies 100% of the requested constraints and provides fair schedules amongst missions with respect to the state-of-the-art for the most oversubscribed weeks of the years 2016 and 2018.

math.OC↗

arXiv2022

Machine Learning Based Relative Orbit Transfer for Swarm Spacecraft Motion Planning

In this paper we describe a machine learning based framework for spacecraft swarm trajectory planning. In particular, we focus on coordinating motions of multi-spacecraft in formation flying through passive relative orbit(PRO) transfers. Accounting for spacecraft dynamics while avoiding collisions between the agents makes spacecraft swarm trajectory planning difficult. Centralized approaches can be used to solve this problem, but are computationally demanding and scale poorly with the number of agents in the swarm. As a result, centralized algorithms are ill-suited for real time trajectory planning on board small spacecraft (e.g. CubeSats) comprising the swarm. In our approach a neural network is used to approximate solutions of a centralized method. The necessary training data is generated using a centralized convex optimization framework through which several instances of the n=10 spacecraft swarm trajectory planning problem are solved. We are interested in answering the following questions which will give insight on the potential utility of deep learning-based approaches to the multi-spacecraft motion planning problem: 1) Can neural networks produce feasible trajectories that satisfy safety constraints (e.g. collision avoidance) and low in fuel cost? 2) Can a neural network trained using n spacecraft data be used to solve problems for spacecraft swarms of differing size?

cs.RO↗

arXiv2021

Optimal Multi-Robot Motion Planning via Parabolic Relaxation

Multi-robot systems offer enhanced capability over their monolithic counterparts, but they come at a cost of increased complexity in coordination. To reduce complexity and to make the problem tractable, multi-robot motion planning (MRMP) methods in the literature adopt de-coupled approaches that sacrifice either optimality or dynamic feasibility. In this paper, we present a convexification method, namely "parabolic relaxation", to generate optimal and dynamically feasible trajectories for MRMP in the coupled joint-space of all robots. We leverage upon the proposed relaxation to tackle the problem complexity and to attain computational tractability for planning over one hundred robots in extremely clustered environments. We take a multi-stage optimization approach that consists of i) mathematically formulating MRMP as a non-convex optimization, ii) lifting the problem into a higher dimensional space, iii) convexifying the problem through the proposed computationally efficient parabolic relaxation, and iv) penalizing with iterative search to ensure feasibility and recovery of feasible and near-optimal solutions to the original problem. Our numerical experiments demonstrate that the proposed approach is capable of generating optimal and dynamically feasible trajectories for challenging motion planning problems with higher success rate than the state-of-the-art, yet remain computationally tractable for over one hundred robots in a highly dense environment.

cs.RO↗

arXiv2021

Rapid Convergence of First-Order Numerical Algorithms via Adaptive Conditioning

This paper is an attempt to remedy the problem of slow convergence for first-order numerical algorithms by proposing an adaptive conditioning heuristic. First, we propose a parallelizable numerical algorithm that is capable of solving large-scale conic optimization problems on distributed platforms such as {graphics processing unit} with orders-of-magnitude time improvement. Proof of global convergence is provided for the proposed algorithm. We argue that on the contrary to common belief, the condition number of the data matrix is not a reliable predictor of convergence speed. In light of this observation, an adaptive conditioning heuristic is proposed which enables higher accuracy compared to other first-order numerical algorithms. Numerical experiments on a wide range of large-scale linear programming and second-order cone programming problems demonstrate the scalability and computational advantages of the proposed algorithm compared to commercial and open-source state-of-the-art solvers.

math.OC↗

arXiv2021

Enhanced Modeling of Contingency Response in Security-constrained Optimal Power Flow

This paper provides an enhanced modeling of the contingency response that collectively reflects high-fidelity physical and operational characteristics of power grids. Integrating active and reactive power contingency responses into the security-constrained optimal power flow (SCOPF) problem is challenging, due to the nonsmoothness and nonconvexity of feasible sets in consequence of piece-wise curves representing generator characteristics. We introduce a continuously-differentiable model using functions that closely resemble PV/PQ switching and the generator contingency response. These models enforce physical and operational limits by optimally allocating active power imbalances among available generators and deciding the bus type to switch from the PV type to the PQ type. The efficacy of this method is numerically validated on the IEEE 30-bus, 300-bus, and 118-bus systems with 12, 10, and 100 contingencies, respectively.

eess.SY↗

arXiv2020

Multi-Agent Motion Planning using Deep Learning for Space Applications

State-of-the-art motion planners cannot scale to a large number of systems. Motion planning for multiple agents is an NP (non-deterministic polynomial-time) hard problem, so the computation time increases exponentially with each addition of agents. This computational demand is a major stumbling block to the motion planner's application to future NASA missions involving the swarm of space vehicles. We applied a deep neural network to transform computationally demanding mathematical motion planning problems into deep learning-based numerical problems. We showed optimal motion trajectories can be accurately replicated using deep learning-based numerical models in several 2D and 3D systems with multiple agents. The deep learning-based numerical model demonstrates superior computational efficiency with plans generated 1000 times faster than the mathematical model counterpart.

cs.RO↗

arXiv2020

Penalized Semidefinite Programming for Quadratically-Constrained Quadratic Optimization

In this paper, we give a new penalized semidefinite programming approach for non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular point that is sufficiently close to the feasible set can be used to construct an appropriate penalty term and recover a feasible solution. Inspired by these results, we develop a heuristic sequential procedure that preserves feasibility and aims to improve the objective value at each iteration. Numerical experiments on large-scale system identification problems as well as benchmark instances from the library of quadratic programming (QPLIB) demonstrate the ability of the proposed penalized semidefinite programs in finding near-optimal solutions for non-convex QCQP.

math.OC↗

arXiv2018

Penalized Parabolic Relaxation for Optimal Power Flow Problem

This paper is concerned with optimal power flow (OPF), which is the problem of optimizing the transmission of electricity in power systems. Our main contributions are as follows: (i) we propose a novel parabolic relaxation, which transforms non-convex OPF problems into convex quadratically-constrained quadratic programs (QCQPs) and can serve as an alternative to the common practice semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, (ii) we propose a penalization technique which is compatible with the SDP, SOCP, and parabolic relaxations and guarantees the recovery of feasible solutions for OPF, under certain assumptions. The proposed penalized convex relaxation can be used sequentially to find feasible and near-globally optimal solutions for challenging instances of OPF. Extensive numerical experiments on small and large-scale benchmark systems corroborate the efficacy of the proposed approach. By solving a few rounds of penalized convex relaxation, fully feasible solutions are obtained for benchmark test cases from [1]-[3] with as many as 13659 buses. In all cases, the solutions obtained are not more than 0.32% worse than the best-known solutions.

math.OC↗

arXiv2018

Sequential Relaxation of Unit Commitment with AC Transmission Constraints

This paper proposes a sequential convex relaxation method for obtaining feasible and near-globally optimal solutions for unit commitment (UC) with AC transmission constraints. First, we develop a second-order cone programming (SOCP) relaxation for AC unit commitment. To ensure that the resulting solutions are feasible for the original non-convex problem, we incorporate penalty terms into the objective of the proposed SOCP relaxation. We generalize our penalization method to a sequential algorithm which starts from an initial point (not necessarily feasible) and leads to feasible and near-optimal solutions for AC unit commitment. Once a feasible point is attained, the algorithm preserves feasibility and improves the objective value until a near optimal point is obtained. The experimental results on IEEE 57, IEEE 118, and IEEE 300 bus benchmark cases from MATPOWER demonstrate the performance of the proposed method in solving challenging instances of AC unit commitment.

math.OC↗

arXiv2018

Convex Relaxation of Bilinear Matrix Inequalities Part I: Theoretical Results

This two-part paper is concerned with the problem of minimizing a linear objective function subject to a bilinear matrix inequality (BMI) constraint. In this part, we first consider a family of convex relaxations which transform BMI optimization problems into polynomial-time solvable surrogates. As an alternative to the state-of-the-art semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, a computationally efficient parabolic relaxation is developed, which relies on convex quadratic constraints only. Next, we developed a family of penalty functions, which can be incorporated into the objective of SDP, SOCP, and parabolic relaxations to facilitate the recovery of feasible points for the original non-convex BMI optimization. Penalty terms can be constructed using any arbitrary initial point. We prove that if the initial point is sufficiently close to the feasible set, then the penalized relaxations are guaranteed to produce feasible points for the original BMI. In Part II of the paper, the efficacy of the proposed penalized convex relaxations is demonstrated on benchmark instances of H2 and Hinf optimal control synthesis problems.

math.OC↗

arXiv2018

Convex Relaxation of Bilinear Matrix Inequalities Part II: Applications to Optimal Control Synthesis

The first part of this paper proposed a family of penalized convex relaxations for solving optimization problems with bilinear matrix inequality (BMI) constraints. In this part, we generalize our approach to a sequential scheme which starts from an arbitrary initial point (feasible or infeasible) and solves a sequence of penalized convex relaxations in order to find feasible and near-optimal solutions for BMI optimization problems. We evaluate the performance of the proposed method on the H2 and Hinfinity optimal controller design problems with both centralized and decentralized structures. The experimental results based on a variety of benchmark control plants demonstrate the promising performance of the proposed approach in comparison with the existing methods.

math.OC↗

arXiv2017

A Scalable Semidefinite Relaxation Approach to Grid Scheduling

Determination of the most economic strategies for supply and transmission of electricity is a daunting computational challenge. Due to theoretical barriers, so-called NP-hardness, the amount of effort to optimize the schedule of generating units and route of power, can grow exponentially with the number of decision variables. Practical approaches to this problem involve legacy approximations and ad-hoc heuristics that may undermine the efficiency and reliability of power system operations, that are ever growing in scale and complexity. Therefore, the development of powerful optimization methods for detailed power system scheduling is critical to the realization of smart grids and has received significant attention recently. In this paper, we propose for the first time a computational method, which is capable of solving large-scale power system scheduling problems with thousands of generating units, while accurately incorporating the nonlinear equations that govern the flow of electricity on the grid. The utilization of this accurate nonlinear model, as opposed to its linear approximations, results in a more efficient and transparent market design, as well as improvements in the reliability of power system operations. We design a polynomial-time solvable third-order semidefinite programming (TSDP) relaxation, with the aim of finding a near globally optimal solution for the original NP-hard problem. The proposed method is demonstrated on the largest available benchmark instances from real-world European grid data, for which provably optimal or near-optimal solutions are obtained.

math.OC↗

arXiv2017

Conic Relaxations for Power System State Estimation with Line Measurements

This paper deals with the non-convex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows, for which existing methods lack guaranteed convergence and theoretical analysis. Motivating by these limitations, we propose a novel convexification framework for the PSSE using semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations. We first study a related power flow (PF) problem as the noiseless counterpart, which is cast as a constrained minimization program by adding a suitably designed objective function. We study the performance of the proposed framework in the case where the set of measurements includes: (i) nodal voltage magnitudes, and (ii) branch active power flows over a spanning tree of the network. It is shown that the SDP and SOCP relaxations both recover the true PF solution as long as the voltage angle difference across each line of the network is not too large (e.g., less than 90 degrees for lossless networks). By capitalizing on this result, penalized SDP and SOCP problems are designed to solve the PSSE, where a penalty based on the weighted least absolute value is incorporated for fitting noisy measurements with possible bad data. Strong theoretical results are derived to quantify the optimal solution of the penalized SDP problem, which is shown to possess a dominant rank-one component formed by lifting the true voltage vector. An upper bound on the estimation error is also derived as a function of the noise power, which decreases exponentially fast as the number of measurements increases.

math.OC↗

arXiv2011

A proposed Optimized Spline Interpolation

The goal of this paper is to design compact support basis spline functions that best approximate a given filter (e.g., an ideal Lowpass filter). The optimum function is found by minimizing the least square problem ($\ell$2 norm of the difference between the desired and the approximated filters) by means of the calculus of variation; more precisely, the introduced splines give optimal filtering properties with respect to their time support interval. Both mathematical analysis and simulation results confirm the superiority of these splines.

cs.MM↗