Interior Point Method Research Articles

Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

A nonsmooth primal-dual method with interwoven PDE constraint solver

We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss–Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

AN OBSERVATIONAL STUDY OF KNOWLEDGE, ATTITUDE AND PRACTICE OF USAGE OF ORAL CONTRACEPTIVE PILLS AMONG UNDERGRADUATE STUDENTS OF THIRD FIRST MBBS IN A TERTIARY CARE TEACHING HOSPITAL

Objective: Education about contraceptive measures in reproductive health is of utmost importance. Doctors can directly influence the patients, and it is, therefore, necessary to know their attitude toward contraceptives. This study will assess the knowledge, attitude, and practice of the usage of oral contraceptive pills (OCPs) among the students of the IIIrd first MBBS. Methods: Among 125 students of the third first MBBS, a questionnaire-based observational knowledge, attitude, perception, and practice study was conducted using Google Forms, at SBKSMI&RC (Shrimati Bhikhiben Kanjibhai Shah Medical Institute and Research Centre), Sumandeep Vidyapeeth in June 2023. Results: In our study, 59.2% of participants were female and 40.8% were male. Students were of the age 20–23 years. 60.8% of students mentioned their source of information as formal education. On being asked about the different uses of OCPs, 69.6% knew about their use as contraception and 52.8% knew their use in menstrual irregularities. In case of a dose being missed, 18.4% did not know what to do whereas 28% would consult a gynecologist and 53.6% knew what to do. 41.6% of students believed that OCPs carry significant side effects. 42.4% preferred getting OCP from medical stores. For usage or prescription in the future, 52.8% preferred barrier methods whereas 24.8% chose intrauterine devices and only 19.2% opted for OCP. When inquired on reasons for not using OCPs, 31.2% opted for fear of side effects. Conclusion: The majority of the students had good knowledge, fair attitude, and fair practice toward the use of OCPs.

365‐day sectional work zone schedule optimization for road networks considering economies of scale and user cost

AbstractThis study proposes a methodology for deriving the optimal work zone schedule for the annual routine maintenance planning in an infrastructure asset management system considering the (i) economies of scale in work zone costs due to work zone synchronization and (ii) user costs across the road network with traffic assignments. A key aspect of the proposed methodology is the ability to derive in detail optimal work zone schedules of realistic‐scale road networks in 100 m sections for 365 days, which is beneficial in practice. To this end, an optimization model of the work zone schedule is newly formulated as a mixed‐integer programming (MIP) problem, and a novel bilevel solution method for the model utilizing the conventional solver Gurobi Optimizer with MIP algorithms such as root relaxation, dual simplex, barrier methods, and cutting planes is proposed. In application examples, the proposed methodology is applied to two real‐world road networks, which confirms that the optimal work zone schedule can be obtained in 313.7 s for a network with 2640, 100 m, sections and in 168,180 s (46 h and 43 min) for a network with 5038, 100 m, sections.

Harder, better, faster, stronger: understanding and improving the tractability of large energy system models

BackgroundEnergy system models based on linear programming have been growing in size with the increasing need to model renewables with high spatial and temporal detail. Larger models lead to high computational requirements. Furthermore, seemingly small changes in a model can lead to drastic differences in runtime. Here, we investigate measures to address this issue.ResultsWe review the mathematical structure of a typical energy system model, and discuss issues of sparsity, degeneracy and large numerical range. We introduce and test a method to automatically scale models to improve numerical range. We test this method as well as tweaks to model formulation and solver preferences, finding that adjustments can have a substantial impact on runtime. In particular, the barrier method without crossover can be very fast, but affects the structure of the resulting optimal solution.ConclusionsWe conclude with a range of recommendations for energy system modellers: first, on large and difficult models, manually select the barrier method or barrier+crossover method. Second, use appropriate units that minimize the model’s numerical range or apply an automatic scaling procedure like the one we introduce here to derive them automatically. Third, be wary of model formulations with cost-free technologies and dummy costs, as those can dramatically worsen the numerical properties of the model. Finally, as a last resort, know the basic solver tolerance settings for your chosen solver and adjust them if necessary.

Flight situational assessment based on complex network and MFIM-TOPSIS

In order to objectively and accurately assess the airborne flight posture in the trajectory-based operation (TBO) mode, a flight posture assessment method based on complex network and MFIM-TOPSIS is proposed. Firstly, the precise protection zone of different aircraft types is established by analyzing the risk of lateral collision between aircraft, and the flight conflict network is constructed by using the three-dimensional velocity barrier method, which is close to the TBO air operation environment, based on which six network topology indexes, such as total node degree, average point strength and network density, are selected to build the flight attitude assessment index system. To weaken the influence of the assessment subjectivity, an interaction matrix that can reflect subjective factors-objective factors is proposed. The maximum information coefficient is used to improve the coding of interaction matrix. Finally the flight attitude level status is assessed by using the superiority-disadvantage solution distance method. The simulation analysis and the example analysis of Takasaki Airport show that the evaluation model can accurately assess the air traffic situation and provide auxiliary information for controllers' decision making and command.

An Enhanced Alternating Direction Method of Multipliers-Based Interior Point Method for Linear and Conic Optimization

An Enhanced Alternating Direction Method of Multipliers-Based Interior Point Method for Linear and Conic Optimization

A hybrid algorithm for quadratically constrained quadratic optimization problems

Quadratically Constrained Quadratic Programs (QCQPs) are an important class of optimization problems with diverse real-world applications. In this work, we propose a variational quantum algorithm for general QCQPs. By encoding the variables in the amplitude of a quantum state, the requirement for the qubit number scales logarithmically with the dimension of the variables, which makes our algorithm suitable for current quantum devices. Using the primal-dual interior-point method in classical optimization, we can deal with general quadratic constraints. Our numerical experiments on typical QCQP problems, including Max-Cut and optimal power flow problems, demonstrate better performance of our hybrid algorithm over classical counterparts.

A decoupled probabilistic constrained topology optimization method based on the constraint shift

AbstractTopology optimization (TO) has recently emerged as an advanced design method. To ensure practical reliability in the design process, it is imperative to incorporate considerations of uncertainty. Consequently, performing reliability analysis (RA) during the design phase becomes necessary. However, RA itself constitutes an optimization problem. Combining these two optimization problems can result in inefficiency. To address this challenge, we propose a decoupled approach that integrates deterministic topology optimization (DTO) and RA cycles. The reliability‐based stress‐constrained TO (RBSCTO) problem is considered in this paper. The DTO constraint is derived based on shifting vectors derived from the previous cycle's RA outcomes, enabling low‐reliability constraint shift towards the feasible direction. The DTO is solved based on solid‐isotropic‐material‐with‐penalization (SIMP) and augmented Lagrangian method. Meanwhile, the optimization problem in RA is addressed using finite differences and the interior point method. To reduce the errors resulting from linear approximation and optimization in RA when the target reliability is very low, an outlier handling method is employed. Meantime, we utilize a probabilistic neural network to enhance the efficiency of reliability assessment. Comparative studies against traditional methods across four RBSCTO tasks are demonstrated to validate its effectiveness. Monte Carlo simulations are used to validate the reliability of results.

In-situ measurement of particle length distribution by FBRM with application to needle- or rod-shape crystallization processes

In this paper, an in-situ measurement method is proposed for detecting needle- or rod-shape particle length distribution (PLD) during a crystallization process, by using the instrument named focused beam reflectance measurement (FBRM). A data pretreatment strategy is firstly presented to modify the measured chord length distribution (CLD) of particles by FBRM for proper analysis, which could guarantee measurement accuracy when the particle length are smaller than 150 μm. Based on the pretreated CLD, a moment conversion algorithm is established for estimating the moments of particle population density, so as to further improve the measurement accuracy for particles possessing larger sizes. Correspondingly, the particle growth-nucleation model parameters are efficiently estimated by using the interior point method (IPM) in combination with the homotopy analysis method (HAM). Then the PLD is reconstructed based on these estimated low-order moments, through a modified cubic-spline interpolation algorithm with variable interpolation grids. Experimental results on the seeded cooling crystallization process of β form L-glutamic acid (LGA) well demonstrate the effectiveness of the proposed method for in-situ measurement.

HIV-Related Knowledge and Health Care Seeking Barriers among Transgender Individuals in a Megacity of Bangladesh: A Cross-sectional Study

The prevalence of HIV among Transgender is 1% in Bangladesh while only 0.9% in Dhaka City.Being minority group barriers faced by transgenders for healthcare-seeking are major public health concerns. Thus, the study aims to find out the HIV-related knowledge and health care seeking barriers among transgender individuals in dhaka city. This was a cross-sectional study purposively selecting 135 transgender from Drop-in-center and three Hijra pollis of Dhaka City. Data was collected using a semi-structured questionnaire containing a Likert scale regarding HIV-related knowledge. Majority were professional sex sellers including 54.10% having irregular sexual clients while 48.9% had been engaged in sex work for more than 10 years. Although more than half (57%) had good HIV-related knowledge. Only one-third (37.30%) sometimes used barrier method and more than half (58%) had knowledge of HIV symptoms. The barriers were negligence (16%), inconvenient location (12.43%), shame (13.45%), judgment (11.68%), fear (7.60%) and long waiting time (8.10%). Transgenders from Drop-in-center had higher HIV-related knowledge compared to ones from hijra pollis [t (95% CI)=.168, p<.001]. Those having HIV-related higher knowledge did not feel the necessity of isolated health corners [t (95% CI)= -3.799, p<.001]. Although DIC provides healthcare services, but existing facilities barriers need to be addressed to extend health services.

Robust Contact Computation in Non-Rigid Variation Simulation

Abstract Geometric variation is an inevitable element of any fabrication process. To secure the geometric quality of the assembled products, variation simulation is performed to control compliance with the set geometric requirements. In non-rigid variation simulation, contact modeling is used to avoid the virtual penetration of the components in the adjacent areas, enhancing the simulation accuracy. For frictionless contact models, numerical errors and convergence issues due to the deformation behavior of the interacting surfaces are still limiting the computational efficiency of solving this optimization problem. The optimization problem associated with a contact model is often large-scale, and in practice, fast and robust methods for achieving convergence are required. Previous implementations of contact modeling for non-rigid variation simulation have been prominently based on the Iterative or Penalty Methods. In this paper, a quadratic programming approach has been introduced, based on the Lagrangian multiplier method, for robust contact modeling in non-rigid variation simulation, and the performance of the proposed approach has been compared to the previously applied Iterative and Interior Point Method. The methods have been compared on three industrial reference cases, and the convergence and time-efficiency of each method are compared. The results show that robust optimization of the quadratic program associated with the contact model is highly dependent on the reduced stiffness matrix condition. Furthermore, it has been shown that robust and efficient contact modeling in non-rigid variation simulation is achievable through the proposed quadratic programming method.

Optimal Approximation of Unique Continuation

AbstractWe consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.

Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions

Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions

Effect of coil and chamber structure on plasma radial uniformity in radio frequency inductively coupled plasma

Enhancing plasma uniformity can be achieved by modifying coil and chamber structures in radio frequency inductively coupled plasma (ICP) to meet the demand for large-area and uniformly distributed plasma in industrial manufacturing. This study utilized a two-dimensional self-consistent fluid model to investigate how different coil configurations and chamber aspect ratios affect the radial uniformity of plasma in radio frequency ICP. The findings indicate that optimizing the radial spacing of the coil enhances plasma uniformity but with a reduction in electron density. Furthermore, optimizing the coil within the ICP reactor, using the interior point method in the Interior Point Optimizer significantly enhances plasma uniformity, elevating it from 56% to 96% within the range of the model sizes. Additionally, when the chamber aspect ratio k changes from 2.8 to 4.7, the plasma distribution changes from a center-high to a saddle-shaped distribution. Moreover, the plasma uniformity becomes worse. Finally, adjusting process parameters, such as increasing source power and gas pressure, can enhance plasma uniformity. These findings contribute to optimizing the etching process by improving plasma radial uniformity.

Large deformation of cable networks with fiber sliding as a second-order cone programming

A new model for irreversible large deformations of fiber networks is developed. The fibers are considered as inextensible cables that slide relative to each other in the frictional junctions. This sliding is constituted by a rate-independent flow rule. The nonsmooth dissipation potential for each sliding system is defined as a product of the yield strength and the absolute value of the fiber sliding. The response of the cable segments is nonsmooth as well, since it shows asymmetry with respect to tension and compression. A principle of minimum incremental potential and a pure complementary energy principle are derived for the equilibrium incremental loading of the network at large deformations. They form a pair of primal and dual second-order cone programming problems with matching sets of displacement-based and force-based variables. These problems can be effectively solved by interior point methods that have many advantages compared to the gradient-based methods or dynamic relaxation. The model is extended by a simple mechanism of fiber pull-out resulting from fiber sliding at the free unconstrained ends. This can be used for the microstructural analysis of failure of needle-punched nonwoven materials.

Contraception after childbirth: current achievements and prospects: A review

The article presents modern methods of contraception after childbirth, including their impact on the mother and child, the main advantages and disadvantages. Special attention is paid to barrier methods of contraception, spermicides, and their advantages according to the "Acceptance Criteria" of the World Health Organization (2015) and the "National Acceptance Criteria" (2023), their features and prospects for use.

Effect of various contraceptives on oocyte yield and maturation in patients undergoing planned oocyte cryopreservation

Research questionDo the various forms of hormonal and non-hormonal contraceptives have any association with ovarian stimulation outcomes, such as oocyte yield and maturation, in patients undergoing planned oocyte cryopreservation (POC)? DesignThis retrospective cohort study included all patients who underwent POC cycles between 2011 and 2023. The use of types of contraception before a POC cycle was recorded. The study evaluated the median number of cumulus–oocyte complexes obtained after vaginal oocyte retrieval and the proportion of metaphase II oocytes that underwent vitrification among all the cohorts. ResultsA total of 4059 oocyte freezing cycles were included in the analysis. Eight types of contraceptive method were recognized in patients undergoing ovarian stimulation: intrauterine device (IUD), copper (n = 84); IUD, levonorgestrel low dose (<52 mg) (n = 37); IUD, levonorgestrel (n = 192); subdermal etonogestrel implant (n = 14); injectable medroxyprogesterone acetate (n = 11); etonogestrel vaginal ring (n = 142); combined oral contraceptive pills (n = 2349); and norelgestromin transdermal patch (n = 10). The control group included patients not using contraceptives or using barrier or calendar methods (n = 1220). Among all the cohorts the median number of cumulus–oocyte complexes retrieved during oocyte retrieval was comparable (P = 0.054), and a significant difference in oocyte maturity rate with median number of vitrified oocytes was found (P = 0.03, P < 0.001, respectively). After adjusting for confounders a multivariate analysis found no association between the type of contraceptive and proportion of metaphase II oocytes available for cryopreservation. ConclusionsAmong the various forms of contraception, none was shown to have an adverse association with oocyte yield or maturation rate in patients undergoing POC.

A convex cone programming based implicit material point method

For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convex cone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convex cone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order cone programming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.

Learning to Optimize Contextually Constrained Problems for Real-Time Decision Generation

The topic of learning to solve optimization problems has received interest from both the operations research and machine learning communities. In this paper, we combine ideas from both fields to address the problem of learning to generate decisions to instances of optimization problems with potentially nonlinear or nonconvex constraints where the feasible set varies with contextual features. We propose a novel framework for training a generative model to produce provably optimal decisions by combining interior point methods and adversarial learning, which we further embed within an iterative data generation algorithm. To this end, we first train a classifier to learn feasibility and then train the generative model to produce optimal decisions to an optimization problem using the classifier as a regularizer. We prove that decisions generated by our model satisfy in-sample and out-of-sample optimality guarantees. Furthermore, the learning models are embedded in an active learning loop in which synthetic instances are iteratively added to the training data; this allows us to progressively generate provably tighter optimal decisions. We investigate case studies in portfolio optimization and personalized treatment design, demonstrating that our approach yields advantages over predict-then-optimize and supervised deep learning techniques, respectively. In particular, our framework is more robust to parameter estimation error compared with the predict-then-optimize paradigm and can better adapt to domain shift as compared with supervised learning models. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2020.03565 .