Dual Interior Point Algorithm Research Articles

Second-order cone programming for frictional contact mechanics using interior point algorithm

We report experiments of an implementation of a primal–dual interior point algorithm for solving mechanical models of one-sided contact problems with Coulomb friction. The objective is to recover an optimal solution with high accuracy and as quickly as possible. These developments are part of the design of Siconos, an open-source software for modelling and simulating non-smooth dynamical systems. Currently, Siconos uses mainly first-order methods for the numerical solution of these systems. These methods are very robust, but suffer from a linear rate of convergence and are therefore too slow to find accurate solutions in a reasonable time. Our main objective is to apply second-order methods to speed up convergence. In this paper, we focus on solving a relaxed formulation of the initial mechanical model, corresponding to the optimality conditions of a convex quadratic minimization problem with second-order cone constraints. We will present in detail a primal–dual interior point algorithm for solving this type of problem. The main difficulty in implementing this algorithm arises from the fact that at each iteration of the algorithm, a change of variable, called a scaling, has to be performed in order to guarantee the non-singularity of the linear system to be solved, as well as to recover a symmetric system. Although this scaling strategy is very nice from a theoretical point of view, it leads to an enormous deterioration of the conditioning of the linear system when approaching the optimal solution and therefore to all the numerical difficulties that result from it. We will describe in detail the numerical algebra that we have developed in our implementation, in order to overcome these problems of numerical instability. We will also present the solution of the models resulting from the problems of rolling friction, for which the cone of constraints is no longer self-dual like the Lorentz cone.

Primal–dual interior-point algorithm for symmetric model predictive control

This paper presents a primal–dual interior-point (pdip) optimization algorithm for solving extreme-scale model predictive control (mpc) problems with linear dynamics, polytopic constraints, and quadratic/linear costs which are all invariant under the symmetric-group. We show that exploiting symmetry can reduce the computational and memory burden of extreme-scale or fast-paced applications of mpc. Our algorithm transforms the original inputs, states, and constraints of the mpc problem into a symmetric domain. The premise of our algorithm is that the numerical linear algebra used to solve the optimization problem has lower computational and memory complexity in the transformed domain. We demonstrate our algorithm for a heating, ventilation, and air-conditioning (hvac) numerical example. We show that, for our largest hvac control problem, our symmetry exploiting the pdip algorithm reduces the computation-time from minutes to seconds in comparison with the baseline pdip algorithm. Furthermore, we show that the presented symmetry exploiting pdip algorithm outperforms a state-of-the-art symmetry exploiting optimization algorithm.

An Algebraic-Based Primal–Dual Interior-Point Algorithm for Rotated Quadratic Cone Optimization

In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

Primal–dual interior-point algorithm for electricity cost minimization in a prosumer-based smart grid environment: A convex optimization approach

This study proposes a primal–dual interior-point algorithm, which is used by the prosumers to efficiently manage the operation of their generating and consuming devices, to maximize their own profits while contributing to the stability of the overall electricity grid. The energy management problem investigated in this study is formulated as a convex optimization problem and the primal–dual interior-point algorithm is leveraged to acquire the optimal solution which minimizes the energy cost for prosumers in a smart grid environment. The proposed algorithm considers the cost of energy generation by each available device or the charge of a utility company, as well as the predicted consumption and production levels of the facility, and develops an operating plan that maximizes the prosumer’s profitability. The cost function for energy generation is convex, as are the constraints that must be satisfied. The primal–dual interior-point algorithm is an efficient algorithm and has attracted attention in the optimization community. The proposed algorithm is evaluated on three prosumer profiles that have different generation and storage characteristics. A variable pricing structure for power supplied or bought by the utility company is considered in this study and a scenario in which prosumers are not allowed to sell to the utility company. It is evident from the simulation results of the MATLAB environment that scenario 1 generates the minimum cost operation profiles compared to scenario 2. The effectiveness and the robustness of the proposed algorithm are demonstrated in the optimal solution obtained in solving the energy management problem.

A dynamic optimization method for power distribution network operation with high ratio photovoltaics

With the high ratio distributed photovoltaics (PVs) penetration to the distribution network (DN), the bearing capacity, integration capacity, consuming capacity and controlling capacity of DN for PVs are all facing great challenges. Enhancing the flexible regulation and operation optimization capabilities of DN have become key factors to satisfy these new challenges. For the operation optimization problem of high ratio PVs penetration, a novel quasi-equal curtailment ratio (QCR) constraint of PVs is proposed in the dynamic optimization method for DN operation. Firstly, the uncertainty of high ratio PVs and their active power deviations are discussed. And a dynamic optimization framework of PVs based on optimal power flow (OPF) has been proposed. Secondly, the optimization model for high ratio PVs is formulated with multi-objective of maximum PVs output, minimum voltage deviation and minimum line loss, under the equality and inequality constraints of power flow, PVs power generation limit, QCR of PVs and so on. Then the multi-objective optimization problem (MOP) is transformed into a single-objective optimization problem (SOP) by weighting coefficients, and solved by sequential quadratic programming (SQP) with trust-region (TR) searching algorithm. Finally, the proposed method is tested, verified and compared with the primal dual interior point (PDIP) and traditional SQP algorithms in IEEE 33-bus, PG&E 69-bus, 292-bus and 1180-bus test cases. The experimental results show the rapidity and robustness of the proposed method.

A novel centroid-enriched edge-based smoothed radial point interpolation method for upper bound limit analysis

This paper presents a novel centroid-enriched edge-based smoothed radial point interpolation method (CE-ESRPIM) to overcome the volumetric locking problem and increase the numerical accuracy for upper bound limit analysis. Based on the edge-based smoothed radial point interpolation method (ESRPIM), the centroid of the triangle element is used as a field node to discretize the problem domain. Simultaneously, the local support node selection scheme, which is established based on a modified centroid-enriched triangle mesh (CE-TScheme), is proposed to realize local radial point interpolation (RPIM). Using the Mohr–Coulomb yield criterion and associated flow rule, the upper bound theorem in the plane strain condition is discretized into a standard second-order cone programming (SOCP) problem through the CE-ESRPIM, which is solved using the primal–dual interior-point algorithm (PDIP). The proposed approach with the enriched centroid considered as the field node is straightforward and efficient and can directly inherit the formulation framework of the traditional ESRPIM. Moreover, this method has been proven to be effective in eliminating the volumetric locking problem under incompressible conditions.

Parameter identification for current–temperature relationship of contact wire under natural convection

The current–temperature relationship of contact wire in high-speed railways is the key to the design of pantograph–catenary system and operational planning of electrified locomotives. The traditional researches on the current–temperature relationship of bare overhead conductors and model parameters mainly focus on aluminum conductor steel reinforced (ACSR), while the description of the heat balance process of contact wire is relatively lacking. In this paper, by optimizing the features of the existing models, an electro-thermal coupling (ETC) model is established to describe the current–temperature relationship under different convection conditions. For the unknown parameters in this model, a parameter identification method based on the nonlinear primal–dual interior point algorithm is applied. The effectiveness of the parameter identification method and the ETC model is verified by the current–temperature experiment of contact wire under different current levels. The experimental results show that the root mean square error of the ETC model using identified parameters is only 1.22%, which demonstrates that the parameter identification method can effectively obtain the model parameters under natural convection. On this basis, a comparison shows that the root mean square error of the ETC model is 6.81% lower than that of the IEEE model, which demonstrates that the ETC model can describe the current–temperature relationship of contact wire under natural convection more accurately after parameter identification.

A primal–dual interior point method for a novel type-2 second order cone optimization

In this paper, we define a new, special second order cone as a type-k second order cone. We focus on the case of k=2, which can be viewed as a second order conic optimization (SOCO) problem with an additional complicating variable. For this new problem, we develop the necessary prerequisites, based on previous work for traditional SOCO problem. We then develop a primal–dual interior point algorithm for solving a type-2 second order conic optimization problem, based on a family of kernel functions suitable for this type-2 SOCO. We finally derive a new iteration bound for our framework.

Collision avoidance maneuver optimization with a multiple-impulse convex formulation

A method to compute optimal collision avoidance maneuvers for short-term encounters is presented. The maneuvers are modeled as multiple-impulses to handle impulsive cases and to approximate finite burn arcs associated either with short alert times or the use of low-thrust propulsion. The maneuver design is formulated as a sequence of convex optimization problems solved in polynomial time by state-of-the-art primal–dual interior-point algorithms. The proposed approach calculates optimal solutions without assumptions about the thrust arc structure and thrust direction. The execution time is fraction of a second for an optimization problem with hundreds of variables and constraints, making it suitable for autonomous calculations.

Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC

We propose a local convergence analysis of a primal–dual interior point algorithm for the solution of a bound-constrained optimization problem. The algorithm includes a regularization technique to prevent singularity of the matrix of the linear system at each iteration, when the second-order sufficient conditions do not hold at the solution. These conditions are replaced by a milder assumption related to a local error-bound condition. This new condition is a generalization of the one used in unconstrained optimization. We show that by an appropriate updating strategy of the barrier parameter and of the regularization parameter, the proposed algorithm owns a superlinear rate of convergence. The analysis is made thanks to a boundedness property of the inverse of the Jacobian matrix arising in interior point algorithms. An illustrative example is given to show the behavior and the gain obtained by this regularization strategy.

Augmented system for detecting the steady‐state voltage stability margin

This work provides an augmented system (AS), constructed based on a modified tangent vector Index, i.e. the full derivative of the uncontrollable parameter with respect to the voltage magnitude of critical bus of the system, to directly detect steady-state voltage stability margin (SSVSM). The Newton iteration produced by the AS is illustrated to be able to approach the equilibrium manifold until converging to the saddle node bifurcation, given that the initial point is approaching to the corresponding branch. With the special searching direction, limit induced dynamic bifurcations or limit induced static bifurcation that appear along the equilibrium manifold are able to be located with expected precision by the proposed corrections. Several cases in a 2000-bus synthetic grid exemplify the effectiveness and high efficiency of the algorithm, compared with standard continuation power flow and the predicter-corrector primal dual interior point algorithm for detecting the SSVSM.

Fast compressed sensing analysis for imaging reconstruction with primal dual interior point algorithm

Compressed sensing (CS) can recover a signal from a small number of observed transforms of that signal. It mainly consists of two complimentary elements including compressed sampling and computational image reconstruction. In this work, we have developed a ghost imaging system and proposed a primal dual interior point compressed sensing algorithm. We also demonstrated the performance of our imaging system and CS algorithm with simulations and experiment compared to conventional approaches. The discrete samples signal can be reconstructed by our CS algorithm. Experimental results show that the proposed compressed imaging method outperforms the conventional CS approaches in both computational time and reconstruction accuracy.

Coordinated optimal dispatch of VPPs in unbalanced ADNs

To manage and control massive distributed energy resources (DERs) integrated into active distribution networks (ADNs), dispersed DERs are aggregated as virtual power plants (VPPs). This study introduces a distributed optimal dispatch strategy of VPPs considering operational constraints in unbalanced ADN. First, a practical multi-VPP coordination optimal dispatch model considering P–Q coupling characteristics in unbalanced ADN is proposed. Then, the primal–dual interior-point algorithm is employed to transform the original model into a Karush–Kuhn–Tucker equation, where afterwards distributed quasi-Newton method is developed to solve the equation. Under the proposed distributed strategy, each VPP only communicates limited boundary information with utility grid. With the advantage of quasi-second-order information, the proposed method outperforms the popular distributed gradient-based algorithm, alternating direction method of multipliers. The distributed method provides merits such as efficient calculation and privacy protection for multi-VPP operation in ADN. Numerical tests evaluate the efficiency and accuracy of the proposed method.

EST image reconstruction based on primal dual interior point algorithm

EST image reconstruction based on primal dual interior point algorithm

An efficient twice parameterized trigonometric kernel function for linear optimization

Recently, Bouafia et al. (J Optim Theory Appl 170:528–545, 2016) investigated a new efficient kernel function that differs from self-regular kernel functions. The kernel function has a trigonometric barrier term. This paper introduces a new efficient twice parametric kernel function that combines the parametric classic function with the parametric kernel function trigonometric barrier term given by Bouafia et al. (J Optim Theory Appl 170:528–545, 2016) to develop primal–dual interior-point algorithms for solving linear programming problems. In addition, we obtain the best complexity bound for large and small-update primal–dual interior point methods. This complexity estimate improves results obtained in Li and Zhang (Oper Res Lett 43(5):471–475, 2015), Peyghami and Hafshejani (Numer Algorithms 67:33–48, 2014) and matches the best bound obtained in Bai et al. (J Glob Optim 54:353–366) and Peng et al. (J Comput Technol 6:61–80, 2001). Finally, our numerical experiments on some test problems confirm that the new kernel function has promising applications compared to the kernel function given by Fathi-Hafshejani (Optimization 67(10):1605–1630, 2018).

Maximum penalized likelihood estimation of additive hazards models with partly interval censoring

Existing likelihood methods for the additive hazards model with interval censored survival data are limited and often ignore the non-negative constraints on hazards. This paper proposes a maximum penalized likelihood method to fit additive hazards models with interval censoring. Our method firstly models the baseline hazard using a finite number of non-negative basis functions, and then regression coefficients and baseline hazard are estimated simultaneously by maximizing a penalized log-likelihood function, where a penalty function is introduced to regularize the baseline hazard estimate. In the estimation procedure, non-negative constraints are imposed on both the baseline hazard and the hazard of each subject. A primal–dual interior-point algorithm is applied to solve the constrained optimization problem. Asymptotic properties are obtained and a simulation study is conducted for assessment of the proposed method.

Efficient Deterministic Algorithm for Huge-Sized Noisy Sensor Localization Problems via Canonical Duality Theory

This paper presents a new deterministic method and a polynomial-time algorithm for solving general huge-sized sensor network localization problems. The problem is first formulated as a nonconvex minimization, which was considered as an NP-hard based on conventional theories. However, by the canonical duality theory, this challenging problem can be equivalently converted into a convex dual problem. By introducing a new optimality measure, a powerful canonical primal-dual interior (CPDI) point algorithm is developed which can solve efficiently huge-sized problems with hundreds of thousands of sensors. The new method is compared with the popular methods in the literature. Results show that the CPDI algorithm is not only faster than the benchmarks but also much more accurate on networks affected by noise on the distances.

Primal–dual interior-point method for linear optimization based on a kernel function with trigonometric growth term

In this paper, we propose a large-update primal–dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature.

A Load-Shedding Model Based on Sensitivity Analysis in on-Line Power System Operation Risk Assessment

The traditional load-shedding models usually use global optimization to get the load-shedding region, which will cause multiple variables, huge computing scale and other problems. This makes it hard to meet the requirements of timeliness in on-line power system operation risk assessment. In order to solve the problems of the present load-shedding models, a load-shedding model based on sensitivity analysis is proposed in this manuscript. By calculating the sensitivity of each branch on each bus, the collection of buses which have remarkable influence on reducing the power flow on over-load branches is obtained. In this way, global optimization is turned to local optimization, which can narrow the solution range. By comprehensively considering the importance of load bus and adjacency principle regarding the electrical coupling relationship, a load-shedding model is established to get the minimum value of the load reduction from different kinds of load buses, which is solved by the primal dual interior point algorithm. In the end, different cases on the IEEE 24-bus, IEEE 300-bus and other multi-node systems are simulated. The correctness and effectiveness of the proposed load-shedding model are demonstrated by the simulation results.

Advances in the simulation of viscoplastic fluid flows using interior-point methods

We present a primal–dual interior point algorithm for the resolution of steady-state viscoplastic fluid flows formulated as a conic optimization problem. We give a complete description of the algorithm including some advanced aspects such as a predictor–corrector and scaling scheme to improve its efficiency. Our interior-point approach is shown to be largely more efficient than Augmented Lagrangian (AL) approaches which are traditionally used to solve such problems. In particular, the interior-point approach is roughly 5 times faster than the modern accelerated version of AL algorithms. The yield surfaces are shown to be accurately predicted and various examples ranging from channel flows to three-dimensional flows through a porous medium demonstrate its efficiency.