We present two parallel optimization algorithms for a convex function $f$. The first algorithm optimizes over linear inequality constraints in a Hilbert space, $\mathbb H$, and the second over a non convex polyhedron in $\mathbb R^n$. The algorithms reduce the inequality constraints to equality constraints, and garner information from subsets of constraints to speed up the process. Let $r \in \mathbb N$ be the number of constraints and $ν(\cdot)$ be the time complexity of some process, then given enough threads, and information gathered earlier from subsets of the given constraints, we compute an optimal point of a polyhedral cone in $O(ν(\langle \cdot,\cdot \rangle) + ν(\min_A f)))$ for affine space $A$, the intersection of the faces of the cone. We then apply the method to all the faces of the polyhedron to find the linear inequality constrained optimum. The methods works on constrained spaces with empty interiors, furthermore no feasible point is required, and the algorithms recognize when the feasible space is empty. The methods iterate over surfaces of the polyhedron and the corresponding affine hulls using information garnered from previous iterations of superspaces to speed up the process.
This paper proposes a parallelizable algorithm for linear-quadratic model predictive control (MPC) problems with state and input constraints. The algorithm itself is based on a parallel MPC scheme that has originally been designed for systems with input constraints. In this context, one contribution of this paper is the construction of time-varying yet separable constraint margins ensuring recursive feasibility and asymptotic stability of sub-optimal parallel MPC in a general setting, which also includes state constraints. Moreover, it is shown how to tradeoff online run-time guarantees versus…
A Newton-type active set algorithm for large-scale minimization subject to polyhedral constraints is proposed. The algorithm consists of a gradient projection step, a second-order Newton-type step in the null space of the constraint matrix, and a set of rules for branching between the two steps. We show that the proposed method asymptotically takes the Newton step when the active constraints are linearly independent and a strong second-order sufficient optimality condition holds. We also show that the method has a quadratic rate of convergence under standard conditions. Numerical experiments a…
A constraint-reduced Mehrotra-Predictor-Corrector algorithm for convex quadratic programming is proposed. (At each iteration, such algorithms use only a subset of the inequality constraints in constructing the search direction, resulting in CPU savings.) The proposed algorithm makes use of a regularization scheme to cater to cases where the reduced constraint matrix is rank deficient. Global and local convergence properties are established under arbitrary working-set selection rules subject to satisfaction of a general condition. A modified active-set identification scheme that fulfills this c…
We present a method for solving the general mixed constrained convex quadratic programming problem using an active set method on the dual problem. The approach is similar to existing active set methods, but we present a new way of solving the linear systems arising in the algorithm. There are two main contributions; we present a new way of factorizing the linear systems, and show how iterative refinement can be used to achieve good accuracy and to solve both types of sub-problems that arise from semi-definite problems.
The edges of a graph are assigned weights and passage times which are assumed to be positive integers. We present a parallel algorithm for finding the shortest path whose total weight is smaller than a pre-determined value. In each step the processing elements are not analyzing the entire graph. Instead they are focusing on a subset of vertices called {\em active vertices}. The set of active vertices at time $t$ is related to the boundary of the ball $B_t$ of radius $t$ in the first passage percolation metric. Although it is believed that the number of active vertices is an order of magnitude …
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proximal linearized augmented Lagrangian algorithm (PLAM) for solving optimization problems with orthogonality constraints. Unlike the classical augmented Lagrangian methods, in our algorithm, the prime variables are updated by minimizing a proximal linear…
In many engineered systems, optimization is used for decision making at time-scales ranging from real-time operation to long-term planning. This process often involves solving similar optimization problems over and over again with slightly modified input parameters, often under tight latency requirements. We consider the problem of using the information available through this repeated solution process to directly learn a model of the optimal solution as a function of the input parameters, thus reducing the need to solve computationally expensive large-scale parametric programs in real time. Ou…
Scheduling flexible sources to promote the integration of renewable generation is one fundamental problem for operating active distribution networks (ADNs). However, existing works are usually based on power flow models, which require network parameters (e.g., topology and line impedance) that may be unavailable in practice. To address this issue, we propose an efficient constraint learning method to operate ADNs. This method first trains multilayer perceptrons (MLPs) based on historical data to learn the mappings from decisions to constraint violations and power loss. Then, power flow constra…
An alternating direction method of multipliers (ADMM) solver is described for optimal resource allocation problems with separable convex quadratic costs and constraints and linear coupling constraints. We describe a parallel implementation of the solver on a graphics processing unit (GPU) using a bespoke quartic function minimizer. An application to robust optimal energy management in hybrid electric vehicles is described, and the results of numerical simulations comparing the computation times of the parallel GPU implementation with those of an equivalent serial implementation are presented.
Bilevel programming can be used to formulate many problems in the field of power systems, such as strategic bidding. However, common reformulations of bilevel problems to mixed-integer linear programs make solving such problems hard, which impedes their implementation in real-life. In this paper, we significantly improve solution speed and tractability by introducing decision trees to learn the active constraints of the lower-level problem, while avoiding to introduce binaries and big-M constants. The application of machine learning reduces the online solving time, by moving the selection of a…
Towards Exact Temporal Aggregation of Time-Coupled Energy Storage Models via Active Constraint Set Identification and Machine Learning
Institute of Electricity Economics and Energy Innovation, Research Center for Energy Economics and Energy Analytics Graz University of Technology（格拉茨工业大学电力经济与能源创新研究所、能源经济与能源分析研究中心）
We present BiqBin, an exact solver for linearly constrained binary quadratic problems. Our approach is based on an exact penalty method to first efficiently transform the original problem into an instance of Max-Cut, and then to solve the Max-Cut problem by a branch-and-bound algorithm. All the main ingredients are carefully developed using new semidefinite programming relaxations obtained by strengthening the existing relaxations with a set of hypermetric inequalities, applying the bundle method as the bounding routine and using new strategies for exploring the branch-and-bound tree. Furtherm…
Real-world problems often involve the optimization of several objectives under multiple constraints. An example is the hyper-parameter tuning problem of machine learning algorithms. In particular, the minimization of the estimation of the generalization error of a deep neural network and at the same time the minimization of its prediction time. We may also consider as a constraint that the deep neural network must be implemented in a chip with an area below some size. Here, both the objectives and the constraint are black boxes, i.e., functions whose analytical expressions are unknown and are …
The second-order cone relaxation of the branch flow model (BFM) and bus injection model (BIM) variants of optimal power flow are well-known to be equivalent for radial networks. In this work we show that in meshed networks with parallel lines, BIM dominates BFM, and propose novel constraints to make them equivalent in general. Furthermore, we develop an improvement to the second-order cone relaxations of optimal power flow, adding novel and valid linear constraints on the lifted current expressions. We develop two simple test cases to highlight the advantages of the proposed constraints. These…
We study scheduling control of parallel processing networks in which some resources need to simultaneously collaborate to perform some activities and some resources multitask. Resource collaboration and multitasking give rise to synchronization constraints in resource scheduling when the resources are not divisible, that is, when the resources cannot be split. The synchronization constraints affect the system performance significantly. For example, because of those constraints, the system capacity can be strictly less than the capacity of the bottleneck resource. Furthermore, the resource sche…
The traditional (active) reliability redundancy allocation problem (RRAP) is used to maximize system reliability by determining the redundancy and reliability variables in each subsystem to satisfy the volume, cost, and weight constraints. The RRAP structure is very simple, that is, redundant components are parallel in each subsystem, and all subsystems are either connected in series or in a bridge network. Owing to its important and practical applications, a novel RRAP, called the general RRAP (GRRAP), is proposed to extend the series-parallel structure or bridge network to a more general net…
Numerical tools for constraints solving are a cornerstone to control verification problems. This is evident by the plethora of research that uses tools like linear and convex programming for the design of control systems. Nevertheless, the capability of linear and convex programming is limited and is not adequate to reason about general nonlinear polynomials constraints that arise naturally in the design of nonlinear systems. This limitation calls for new solvers that are capable of utilizing the power of linear and convex programming to reason about general multivariate polynomials. In this p…
Active Constraint Learning in High Dimensions from Demonstrations
School of Mechanical Engineering Georgia Institute of Technology（机械工程学院佐治亚理工学院）；School of Electrical and Computer Engineering Georgia Institute of Technology（电气与计算机工程学院佐治亚理工学院）；Schools of Cybersecurity & Privacy and Aerospace Engineering Georgia Institute of Technology（网络安全与隐私学院及航空航天工程学院佐治亚理工学院）
