Convex Programming Research Articles

A Tighter Relaxation for the Relative Pose Problem Between Cameras

This paper tackles the resolution of the Relative Pose problem with optimality guarantees by stating it as an optimization problem over the set of essential matrices that minimizes the squared epipolar error. We relax this non-convex problem with its Shor’s relaxation, a convex program that can be solved by off-the-shelf tools. We follow the empirical observation that redundant but independent constraints tighten the relaxation. For that, we leverage equivalent definitions of the set of essential matrices based on the translation vectors between the cameras. Overconstrained characterizations of the set of essential matrices are derived by the combination of these definitions. Through extensive experiments on synthetic and real data, our proposal is empirically proved to remain tight and to require only 7 milliseconds to be solved even for the overconstrained formulations, finding the optimal solution under a wide variety of configurations, including highly noisy data and outliers. The solver cannot certify the solution only in very extreme cases, e.g.noise 100~{texttt {pix}} and number of pair-wise correspondences under 15. The proposal is thus faster than other overconstrained formulations while being faster than the minimal ones, making it suitable for real-world applications that require optimality certification.

Pacing Equilibrium in First Price Auction Markets

Mature internet advertising platforms offer high-level campaign management tools to help advertisers run their campaigns, often abstracting away the intricacies of how each ad is placed and focusing on aggregate metrics of interest to advertisers. On such platforms, advertisers often participate in auctions through a proxy bidder, so the standard incentive analyses that are common in the literature do not apply directly. In this paper, we take the perspective of a budget management system that surfaces aggregated incentives—instead of individual auctions—and compare first and second price auctions. We show that theory offers surprising endorsement for using a first price auction to sell individual impressions. In particular, first price auctions guarantee uniqueness of the steady-state equilibrium of the budget management system, monotonicity, and other desirable properties, as well as efficient computation through the solution to the well-studied Eisenberg–Gale convex program. Contrary to what one can expect from first price auctions, we show that incentives issues are not a barrier that undermines the system. Using realistic instances generated from data collected at real-world auction platforms, we show that bidders have small regret with respect to their optimal ex post strategy, and they do not have a big incentive to misreport when they can influence equilibria directly by giving inputs strategically. Finally, budget-constrained bidders, who have significant prevalence in real-world platforms, tend to have smaller regrets. Our computations indicate that bidder budgets, pacing multipliers, and regrets all have a positive association in statistical terms. This paper was accepted by Gabriel Weintraub, revenue management and market analytics. Funding: D. Panigrahi was supported in part by the National Science Foundation [Awards CCF 1535972, CCF 1750140, and CCF 1955703]. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2022.4310 .

An Alternating Iteration Algorithm for a Parameter-Dependent Distributionally Robust Optimization Model

Based on a successive convex programming method, an alternating iteration algorithm is proposed for solving a parameter-dependent distributionally robust optimization. Under the Slater-type condition, the convergence analysis of the algorithm is obtained. When the objective function is convex, a modified algorithm is proposed and a less-conservative solution is obtained. Lastly, some numerical tests results are illustrated to show the efficiency of the algorithm.

Reliability Enhancement for VR Delivery in Mobile-Edge Empowered Dual-Connectivity Sub-6 GHz and mmWave HetNets

The reliability of current virtual reality (VR) delivery is low due to the limited resources on VR head-mounted displays (HMDs) and the transmission rate bottleneck of sub-6 GHz networks. In this paper, we propose a dual-connectivity sub-6 GHz and mmWave heterogeneous network architecture empowered by mobile edge capability. The core idea of the proposed architecture is to utilize the complementary advantages of sub-6 GHz links and mmWave links to conduct a collaborative edge resource design, which aims to improve the reliability of VR delivery. From the perspective of stochastic geometry, we analyze the reliability of VR delivery and theoretically demonstrate that sub-6 GHz links can be used to enhance the reliability of VR delivery despite the large mmWave bandwidth. Based on our analytical work, we formulate a joint caching and computing optimization problem with the goal to maximize the reliability of VR delivery. By analyzing the coupling caching and computing strategies at HMDs, sub-6 GHz and mmWave base stations (BSs), we further transform the problem into a multiple-choice multi-dimension knapsack problem. A best-first branch and bound algorithm and a difference of convex programming algorithm are proposed to obtain the optimal and sub-optimal solution, respectively. Numerical simulations demonstrate the performance improvement using the proposed algorithms, and reveal that caching more monocular videos at sub-6 GHz BSs and more stereoscopic videos at mmWave BSs can improve the VR delivery reliability efficiently.

Optimization-Based Trajectory Tracking Approach for Multi-Rotor Aerial Vehicles in Unknown Environments

The goal of this paper is to develop a continuous optimization-based refinement of the reference trajectory to ‘push it out’ of the obstacle-occupied space in the global phase for Multi-rotor Aerial Vehicles in unknown environments. Our proposed approach comprises two planners: a global planner and a local planner. The global planner refines the initial reference trajectory when the trajectory goes either through an obstacle or near an obstacle and lets the local planner calculate a near-optimal control policy. The global planner comprises two convex programming approaches: the first one helps to refine the reference trajectory, and the second one helps to recover the reference trajectory if the first approach fails to refine. The global planner mainly focuses on real-time performance and obstacles avoidance, whereas the proposed formulation of the constrained nonlinear model predictive control-based local planner ensures safety, dynamic feasibility, and the reference trajectory tracking accuracy for low-speed maneuvers, provided that local and global planners have mean computation times 0.06 s (15 Hz) and 0.05 s (20 Hz), respectively, on an NVIDIA Jetson Xavier NX computer. The results of our experiment confirmed that, in cluttered environments, the proposed approach outperformed three other approaches: sampling-based pathfinding followed by trajectory generation, a local planner, and graph-based pathfinding followed by trajectory generation.

Solving certain complementarity problems in power markets via convex programming

Solving certain complementarity problems in power markets via convex programming

Estimation of dynamic systems using a method of characteristics filter

This paper deals with the technical formulation and implementation details of an algorithm that provides an estimate of the probability density function of the state of a nonlinear dynamical system from discrete time measurements. In contrast to other state estimation methods that propagate the conditional moments, the proposed filter is shown to propagate and update the full probability density function of the state. Characteristic solutions to the Liouville equation are used to propagate the exact probability density values along the flow of the dynamical system. The reconstruction of the state probability density function is then posed as a convex programming problem. An adaptive regression-tree based region-decomposition approach is used to efficiently compute expectation integrals involved in implementing the Bayes rule associated with posterior density function. Numerical examples capturing the non-Gaussian nature of the uncertainty in the duffing oscillator problem and the two body problem are used to demonstrate the efficacy of the proposed methods.

Multiscale heterogeneous optimal lockdown control for COVID-19 using geographic information

We study the problem of synthesizing lockdown policies—schedules of maximum capacities for different types of activity sites—to minimize the number of deceased individuals due to a pandemic within a given metropolitan statistical area (MSA) while controlling the severity of the imposed lockdown. To synthesize and evaluate lockdown policies, we develop a multiscale susceptible, infected, recovered, and deceased model that partitions a given MSA into geographic subregions, and that incorporates data on the behaviors of the populations of these subregions. This modeling approach allows for the analysis of heterogeneous lockdown policies that vary across the different types of activity sites within each subregion of the MSA. We formulate the synthesis of optimal lockdown policies as a nonconvex optimization problem and we develop an iterative algorithm that addresses this nonconvexity through sequential convex programming. We empirically demonstrate the effectiveness of the developed approach by applying it to six of the largest MSAs in the United States. The developed heterogeneous lockdown policies not only reduce the number of deceased individuals by up to 45 percent over a 100 day period in comparison with three baseline lockdown policies that are less heterogeneous, but they also impose lockdowns that are less severe.

A new projection neural network for linear and convex quadratic second-order cone programming

A new projection neural network approach is presented for the linear and convex quadratic second-order cone programming. In the method, the optimal conditions of the linear and convex second-order cone programming are equivalent to the cone projection equations. A Lyapunov function is given based on the G-norm distance function. Based on the cone projection function, the descent direction of Lyapunov function is used to design the new projection neural network. For the proposed neural network, we give the Lyapunov stability analysis and prove the global convergence. Finally, some numerical examples and two kinds of grasping force optimization problems are used to test the efficiency of the proposed neural network. The simulation results show that the proposed neural network is efficient for solving some linear and convex quadratic second-order cone programming problems. Especially, the proposed neural network can overcome the oscillating trajectory of the exist projection neural network for some linear second-order cone programming examples and the min-max grasping force optimization problem.

Yosida-regularization based differential equation approach to generalized equations with applications to nonlinear convex programming

Yosida-regularization based differential equation approach to generalized equations with applications to nonlinear convex programming

Uncertain trajectory planning integrating polynomial chaos and convex programming

The present work explores the robust trajectory optimization scheme considering both the initial state disturbance and multiple constraints. An uncertain multi-constraint optimization model has been first established. Due to the existence of the stochastic disturbance, standard numerical trajectory planning algorithms cannot be directly applied to address the considered issue. Hence, based on the intrusive polynomial chaos expansion, we present a deterministic quantification for the stochastic state and constraints, so that the transformed optimization model becomes solvable for standard numerical optimization methods. To obtain the enhanced computational performance, an hp pseudo-spectral sequential convex programming procedure combined with a penalty function and backtracking search is proposed. This is achieved by discretizing and convexifying the nonlinear dynamics/constraints using hp quadrature collocation and successive linearization, respectively, and by adjusting the confidence region manually in the iteration. The simulation of a three-dimensional interception with the specific impact angle is conducted to verify the effectiveness. The simulation results show that the initial solutions are insensitive to the convex optimization, and the control commands generated by the proposed algorithm are effective against the initial state disturbance.

User power allocation and relay beamforming design for secrecy sum rate maximization in two-way relay networks

In this paper, we investigate the physical-layer security (PLS) for multiple peer-to-peer communications in a two-way relaying networks involving 2 L single-antenna users, an N R -antenna relay, and an N E -antenna eavesdropper. Only a pair of peer-to-peer users demand security to exchange messages because the eavesdropper intends to wiretap them in both transmission phases. Our goal is to maximize the achievable secrecy sum rate (SSR) of specific peer-to-peer users, provided that restrictive constraints are satisfied. We endorse a null space beamforming technique at the relay to simplify the optimization problem and exclude private information leakage. The main problem is still very complicated due to coupled variables of the user transmit power with the relay beamforming matrix. Therefore, we convert the main problem into two subproblems, one in terms of the relay beamforming matrix and the other in terms of the user transmit power. To find a locally optimal solution for the relay beamforming, we turn the main problem into a difference of convex (DC) programming problem and for user power allocation, we drive the main problem into a constrained nonlinear optimization problem. Finally, simulations are performed to evaluate the performance benefits of the proposed method.

Multi-tenant resource sharing with equitable-priority-based performance isolation of slices for 5G cellular systems

Network slicing is a promising technique to enable multi-tenant operation in fifth generation (5G) cellular systems. However, efficient implementation of this technique at the air interface requires an adequate resource allocation policy that provides slice isolation and takes into account the heterogeneity of services and their performance requirements. We propose a new slicing scheme aimed at slice performance isolation, efficient capacity utilization, and fair resource allocation among all users. The policy ensures a slice-specific minimum data rate to all slice users as long as their number remains within a contracted limit, yet provides higher data rates whenever capacity is available. Specifically, we incorporate the best features of the complete partitioning and complete sharing policies, while ensuring flexibility and customization for network operators. The proposed scheme is based on an iterative solution of a convex programming problem characterized by polynomial complexity, which makes it suitable for on-line implementation. Associated algorithms are also proposed. Our numerical results demonstrate that the proposed slicing scheme is capable to accommodate heterogeneous traffic and provides performance isolation of slices while maintaining resource utilization at the level of complete sharing. Depending on the system parameters, the proposed scheme allows to improve session loss probability by an order of magnitude as compared to static slicing, while keeping the user data rates comparable to that of the complete sharing scheme and improving the average user satisfaction index by up to 90%. Overall, the proposed scheme results in efficient resource utilization compared to slicing policies in which slice isolation is provided via reservation by setting fixed upper or lower capacity bounds.

Modeling and evaluating the impact of electricity price on commute network flows of battery electric vehicles

ABSTRACT In many urban areas and urban-rural regions, the price of electricity is not only subject to time-of-use tariffs, but also dependent on the usage purpose or service sector, significantly varying across different geographic locations or land uses. Such a spatial price difference greatly affects the charging behavior of electric vehicle drivers. This paper models and analyzes the possible impact of electricity-charging price on the routing choice and flow pattern of electric vehicles in a home-workplace commute traffic network. The particular circumstance in such a network accommodating electric vehicles is that commuters driving battery electric vehicles have opportunities to charge their vehicles at both home and workplace and take into account different electricity-charging prices in their travel choices. For the illustration and evaluation purposes, we presented a small synthetic example and proposed a convex programming model for characterizing the commute network flow pattern of electric vehicles that may be charged at home and/or workplace, encapsulating a piecewise electricity-charging cost function that implies commuters’ combined home-based and workplace-based charging behaviors. Given different electricity-charging prices at home and workplace, solving this problem requires tracking all individual paths in the network. As such, a path-based solution algorithm sketched in the disaggregated simplicial decomposition framework, with some modifications in the path generation phase and an added k-shortest path search procedure, is designed for problem solutions. We implemented the modeling and solution methods for deriving the network flow pattern of electric vehicles and comparing it with that of gasoline vehicles, as well as evaluating the network performance change caused by varying electricity-charging prices with multiple network-level and link-level evaluation matrices.

New Inexact Parallel Splitting Algorithm for a Class of Separable Convex Programming

In this paper, a new inexact parallel splitting algorithm based on the shrinkage algorithm is proposed. We bring in inexact terms, and new iteration points are obtained by the parallel splitting method. A new descent direction and a proper step length are derived. Under reasonable assumptions, the convergence of the algorithm is demonstrated. Some matrix correction problem experiments show that the algorithm is efficient and easy to implement.

Formula omitted]-Kernel-free soft quadratic surface support vector regression

In this paper, we propose a new regression method called the ∊-kernel-free soft quadratic surface support vector regression (∊-SQSSVR). After converting the n-dimensional regression problem into the (n+1)-dimensional classification problem, the principle of maximizing the sum of relative geometrical margin of each training point is used to construct our optimization problem, where the quadratic surface is restricted to be a hyperparaboloid by setting both the (n+1)-th row and (n+1)-th column of the corresponding matrix to be zero. The existence and uniqueness of the optimal solution to both primal and dual problems are also addressed. It should be pointed out that our model is nonlinear and kernel-free, so it does not need to select kernel function and corresponding parameters. At the same time, it is highly interpretable. In addition, our model is still a quadratic convex programming problem similar to the standard SVR. To visualize the effectiveness of our ∊-SQSSVR, 6 artificial datasets and 15 benchmark datasets are implemented in numerical experiments. The results show that our method is less time-consuming and as good as the nonlinear standard SVR with kernel function in comprehensive performances.

Chance-Constrained Covariance Steering in a Gaussian Random Field via Successive Convex Programming

The problem of optimizing affine feedback laws that explicitly steer the mean and covariance of an uncertain system state in the presence of a Gaussian random field is considered. Spatially dependent disturbances are successively approximated with respect to a nominal trajectory by a sequence of jointly Gaussian random vectors. Sequential updates to the nominal control inputs are computed via convex optimization that includes the effect of affine state feedback, the perturbing effects of spatial disturbances, and chance constraints on the closed-loop state and control. The developed method is applied to solve for an affine feedback law to minimize the 99th percentile of required to complete an aerocapture mission around a planet with a randomly disturbed atmosphere.

Optimal vault problem – form finding through 2D convex program

This work puts forward a form finding problem of designing a least-volume vault that is a surface structure spanning over a plane region, which via pure compression transfers a vertically tracking load to the supporting boundary. Through a duality scheme, developed recently for the design of pre-stressed membranes, the optimal vault problem is reduced to a pair of mutually dual convex problems formulated on the 2D reference region. The vault constructed upon solutions of those problems is both of minimum volume and minimum compliance; analytical examples of optimal vaults are given. Through a measure-theoretic approach, such optimal vaults are proved to solve the Prager problem of designing a 3D structure that by compression carries a transmissible load. The ground structure method applied to the convex problems furnishes a pair of discrete, conic quadratic programs leading to optimal design of grid-shells. By adopting the member-adding adaptive technique, this pair is efficiently tackled numerically. The computer method is demonstrated on a number of examples where highly precise grid-shell approximations of optimal vaults are found.

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

An Interior Point Parameterized Central Path Following Algorithm for Linearly Constrained Convex Programming

An Interior Point Parameterized Central Path Following Algorithm for Linearly Constrained Convex Programming