Disciplined multi-convex programming

Shen, Xinyue; Diamond, Steven; Udell, Madeleine; Gu, Yuantao; Boyd, Stephen

doi:10.1109/ccdc.2017.7978647

Cited by 37 publications

(19 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Problem P2 r is a convex problem for either variable x or b, separately, but not jointly. Hence, problem P2 r can be solved by using multi-convex programming by alternatively updating b and x using the modified algorithm described below [21]. The proximal terms are added in the cost function (see below) where the convergence of this algorithm is proved in [22].…”

Section: Solution Methods a General Casementioning

confidence: 99%

Performance Specified State Estimation With Minimum Risk

Aghapour¹,

Farrell²

2018

2018 Annual American Control Conference (ACC)

View full text Add to dashboard Cite

Measurements that significantly deviate from those predicted by the model or from the normal pattern of sensed data are considered as outliers. Since outliers can degrade the performance of state estimation, outlier accommodation is critical. The traditional Neyman-Pearson Kalman filter approach is to ignore all residuals greater than a designer specified threshold. The criticism of such techniques is that they allow missed detections to pass through undetected thereby corrupting both the state estimate and covariance. This causes the state estimation gain and all subsequent outlier decisions to be based on an invalid model. In sensor rich applications, where large numbers of sensor measurements are available, all the sensor data may not be needed to achieve the system accuracy specification. Global navigation satellite systems (GNSS) is one such sensor rich application since various satellite systems are available, each of which supplies more than the minimal number of satellites needed to estimate the system state. Using more data than is required to meet the specification exposes the state estimate to unneeded risk of outlier inclusion. This paper formulates and solves the state estimation problem from the perspective of minimizing risk while achieving a performance specification.

show abstract

Section: Solution Methods a General Casementioning

confidence: 99%

Performance Specified State Estimation With Minimum Risk

Aghapour¹,

Farrell²

2018

2018 Annual American Control Conference (ACC)

View full text Add to dashboard Cite

show abstract

“…Term set 3 is not convex, rendering the whole objective formulation non‐convex. Fortunately, the formulation is multiconvex—by updating one variable while holding the other two constant, we have a convex module. Similar to a previous study, the algorithm is broken down into three modules, which are evaluated in an alternating block fashion.…”

Section: Methodsmentioning

confidence: 99%

“…In this study, we integrate the multiphase piecewise constant Mumford‐Shah function with the fluence map optimization problem into a multiconvex formulation—a non‐convex problem that yields a convex subproblem when all but one block of variables are held constant. This is commonly evaluated by an alternating module scheme.…”

Section: Introductionmentioning

confidence: 99%

Deterministic direct aperture optimization using multiphase piecewise constant segmentation

et al. 2017

View full text Add to dashboard Cite

Purpose Direct Aperture Optimization (DAO) attempts to incorporate machine constraints in the inverse optimization to eliminate the post-processing steps in fluence map optimization (FMO) that degrade plan quality. Current commercial DAO methods utilize a stochastic or greedy approach to search a small aperture solution space. In this study, we propose a novel deterministic direct aperture optimization that integrates the segmentation of fluence map in the optimization problem using the multiphase piecewise constant Mumford-Shah formulation. Methods The Mumford-Shah based direct aperture optimization problem was formulated to include an L2-norm dose fidelity term to penalize differences between the projected dose and the prescribed dose, an anisotropic total variation term to promote piecewise continuity in the fluence maps, and the multiphase piecewise constant Mumford-Shah function to partition the fluence into pairwise discrete segments. A proximal-class, first-order primal-dual solver was implemented to solve the large scale optimization problem, and an alternating module strategy was implemented to update fluence and delivery segments. Three patients of varying complexity—one glioblastoma multiforme (GBM) patient, one lung (LNG) patient, and one bilateral head and neck (H&N) patient with 3 PTVs—were selected to test the new DAO method. For each patient, 20 non-coplanar beams were first selected using column generation, followed by the Mumford-Shah based DAO (DAOMS). For comparison, a popular and successful approach to DAO known as simulated annealing—a stochastic approach—was replicated. The simulated annealing DAO (DAOSA) plans were then created using the same beam angles and maximum number of segments per beam. PTV coverage, PTV homogeneity true(D95D5true), and OAR sparing were assessed for each plan. In addition, high dose spillage, defined as the 50% isodose volume divided by the tumor volume, as well as conformity, defined as the van’t Riet conformation number, were evaluated. Results DAOMS achieved essentially the same OAR doses compared with the DAOSA plans for the GBM case. The average difference of OAR Dmax and Dmean between the two plans were within 0.05% of the plan prescription dose. The lung case showed slightly improved critical structure sparing using the DAOMS approach, where the average OAR Dmax and Dmean were reduced by 3.67% and 1.08%, respectively, of the prescription dose. The DAOMS plan substantially improved OAR dose sparing for the H&N patient, where the average OAR Dmax and Dmean were reduced by over 10% of the prescription dose. The DAOMS and DAOSA plans were comparable for the GBM and LNG PTV coverage, while the DAOMS plan substantially improved the H&N PTV coverage, increasing D99 by 6.98% of the prescription dose. For the GBM and LNG patients, the DAOMS and DAOSA plans had comparable high dose spillage but slightly worse conformity with the DAOMS approach. For the H&N plan, DAOMS was considerably superior in high dose spillage and conformity to the DAOSA. The deterministic...

show abstract

“…The constraints in (14)- (18) are LMIs in the variables K and α. Finding a feasible solution of a set of LMIs can be done by solving an SDP.…”

Section: Policy Synthesis Via Convex Optimizationmentioning

confidence: 99%

“…The sufficient condition based on semidefinite programming involves searching for a diagonal Lyapunov function that guarantees the stability. As it is only a sufficient condition, we propose another approach based on coordinate descent [17], [18]. In each step, we update the variables with in the coordinate descent method to improve the convergence.…”

Section: Introductionmentioning

confidence: 99%

Switched Linear Systems Meet Markov Decision Processes: Stability Guaranteed Policy Synthesis

Cubuktepe

Topcu

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

Switched linear systems are time-varying nonlinear systems whose dynamics switch between different modes, where each mode corresponds to different linear dynamics. They arise naturally to model unexpected failures, environment uncertainties or system noises during system operation. In this paper, we consider a special class of switched linear systems where the mode switches are governed by Markov decision processes (MDPs). We study the problem of synthesizing a policy in an MDP that stabilizes the switched system. Given a policy, the switched linear system becomes a Markov jump linear system whose stability conditions have been extensively studied. We make use of these stability conditions and propose three different computation approaches to find the stabilizing policy in an MDP. We derive our first approach by extending the stability conditions for a Markov jump linear system to handle switches governed by an MDP. This approach requires finding a feasible solution of a set of bilinear matrix inequalities, which makes policy synthesis typically challenging. To improve scalability, we provide two approaches based on convex optimization. We give three examples to show and compare our proposed solutions.

show abstract

Disciplined multi-convex programming

Cited by 37 publications

References 28 publications

Performance Specified State Estimation With Minimum Risk

Performance Specified State Estimation With Minimum Risk

Deterministic direct aperture optimization using multiphase piecewise constant segmentation

Switched Linear Systems Meet Markov Decision Processes: Stability Guaranteed Policy Synthesis

Contact Info

Product

Resources

About