Convex Nonlinear Programming Research Articles

Higher-order interior point methods for convex nonlinear programming

This paper extends the concept of higher-order search directions within interior point methods to convex nonlinear programming. This includes the mathematical framework needed to compute the higher-order derivatives. The paper also highlights some special cases where the computation of these higher-order derivatives is simplified and a dimensional lifting procedure for transforming a large number of general nonlinear problems into one of these more efficient forms. The paper further describes the algorithmic development required to employ these higher-order search directions in a practical algorithm. Computational results are presented for a large number of test problems, highlighting higher-order methods’ strong potential for decreasing iteration count and their case-by-case potential for decreasing CPU time.

Utilizing a projection neural network to convex quadratic multi‐objective programming problems

SummaryIn this paper, we propose a projection neural network model for solving convex quadratic multi‐objective optimization problem (CQMOP). The CQMOP is first converted into an equivalent convex nonlinear programming problem by the means of the weighted sum method, where the Pareto optimal solutions are calculated via different values of weights. A neural network model is then constructed for solving the obtained convex problem. It is shown that the presented neural network is stable in the sense of Lyapunov and is globally convergent. Simulation results are given to illustrate the global convergence and performance of the suggested model. Both theoretical and numerical approaches are studied. Numerical results are in good agreement with the proved theoretical concepts.

Distributionally robust expected residual minimization for stochastic variational inequality problems

The stochastic variational inequality problem (SVIP) is an equilibrium model that includes random variables and has been widely applied in various fields such as economics and engineering. Expected residual minimization (ERM) is an established model for obtaining a reasonable solution for the SVIP, and its objective function is an expected value of a suitable merit function for the SVIP. However, the ERM is restricted to the case where the distribution is known in advance. We extend the ERM to ensure the attainment of robust solutions for the SVIP under the uncertainty distribution (the extended ERM is referred to as distributionally robust expected residual minimization (DRERM), where the worst-case distribution is derived from the set of probability measures in which the expected value and variance take the same sample mean and variance, respectively). Under suitable assumptions, we demonstrate that the DRERM can be reformulated as a deterministic convex nonlinear semidefinite programming to avoid numerical integration.

General-purpose preconditioning for regularized interior point methods

In this paper we present general-purpose preconditioners for regularized augmented systems, and their corresponding normal equations, arising from optimization problems. We discuss positive definite preconditioners, suitable for CG and MINRES. We consider “sparsifications" which avoid situations in which eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems arising from the application of regularized interior point methods to linear or nonlinear convex programming problems.

Robust Interior Point Method for Quantum Key Distribution Rate Computation

Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.

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

Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming

Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.

A Barzilai and Borwein regularization feasible direction algorithm for convex nonlinear SOC programming with linear constraints

A Barzilai and Borwein regularization feasible direction algorithm is proposed for convex nonlinear second-order cone programming with linear constraints. In the algorithm, a regularization penalty term based on the Barzilai and Borwein parameters is added to the Frank–Wolfe linearization objective function of the direction generating subproblem. The Barzilai and Borwein subproblem is transformed into a multi-block separable convex quadratic second-order cone programming (CQSOCP) subproblem. A parallel inexact alternating direction method is applied to solve the multi-block separable CQSOCP subproblem. The global convergence is given. Numerical results demonstrate that our method is efficient for some random convex nonlinear second-order cone programming problems with low accuracy.

Оптимізаційні задачі модернізації пропускних здатностей дуг відмовостійких мереж

Розглянуто математичні моделі двох класів задач модернізації пропускних здатностей дуг відмовостійких орієнтованих мереж. Відмовостійкою вважається мережа, для якої можна задовольнити всі вимоги на передачу потоків, якщо матиме місце одна, будь-яка відмова, з усіх можливих одиничних відмов мережі. У першому класі задач (задача A) для передачі потоків можуть використовуватись всі можливі шляхи в мережі. У другому класі задач (задача P) для передачі потоків задіяно тільки шляхи із напередзаданої множини шляхів. Математичні моделі представлено задачами лінійного, булевого та нелінійного програмування з блочною структурою матриці обмежень. Матеріал статті представлено в п’яти розділах. У розд. 1 описано поняття одиничної відмови та сценарію відмов мережі, наведено зміст оптимізаційних задач A та P для модернізації пропускних здатностей дуг відмовостійкої мережі, описано тестову мережу (6 вершин та 19 дуг) для перевірки алгоритмів розв’язання задач модернізації відмовостійких мереж. У розд. 2 описано базові моделі задач лінійного програмування для знаходження пропускних здатностей дуг відмовостійкої фізичної структури мережі (задача A) та відмовостійкої логічної структури мережі (задача P), розглянуто їх властивості. У розд. 3 описано задачі A та P у формі моделей змішаного булевого лінійного програмування. Наведено оптимальні розв’язки задачі A для різних сценаріїв відмов на прикладі тестової мережі. Розв’язки знайдено за допомогою програми Gurobi з NEOS-сервера, де математичну модель задачі A описано мовою моделювання AMPL. У розд. 4 описано нелінійні моделі опуклого програмування для задач A та P, призначені для знаходження оптимальних за вибраним критерієм пропускних здатностей дуг відмовостійких мереж, та описано декомпозиційний алгоритм їх розв’язання. У розд. 5 наведено опис програмного забезпечення мовою програмування ФОРТРАН для декомпозиційного алгоритму на основі ефективних реалізацій r-алгоритмів Шора. Проведено порівняння декомпозиційного алгоритму з програмою IPOPT на основі результатів розв’язання тестових задач.

A subgradient proximal method for solving a class of monotone multivalued variational inequality problems

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

Ship's response to low-sulfur regulations: From the perspective of route, speed and refueling strategy

Optimizing speed, routing, and refueling strategy are three ways to reduce ship operating costs without modifying on ship. Most past studies have simultaneously optimized one or two variables, resulting in a local optimal solution. To limit sulfur emissions, some governments have established Emission Control Areas (ECAs) in recent years, where ships must use low sulphur fuels. The constraints created by ECAs makes the simultaneous optimization of the three variables even more challenging. This study developed a nonlinear mixed integer programming model to address the challenge of simultaneously optimizing the three variables. After transforming the nonlinear discrete mixed integer programming model into a convex nonlinear continuous programming model, the study proposed a solution algorithm for the proposed model. The new model in this study was applied to the Europe-United States (US) and Southeast Asia service route. The results show that the optimal refueling policy, route, and speed selection can be determined simultaneously using the proposed model and solution method. The study investigated the impact of fuel price on ship refueling strategy, route, and speed; and developed insights from the example analyses.

Joint optimization of computing ratio and access points\u2019 density for mixed mobile edge/cloud computing

Cooperation between the mobile edge computing (MEC) and the mobile cloud computing (MCC) in offloading computing could improve quality of service (QoS) of user equipments (UEs) with computation-intensive tasks. In this paper, in order to minimize the expect charge, we focus on the problem of how to offload the computation-intensive task from the resource-scarce UE to access point’s (AP) and the cloud, and the density allocation of APs’ at mobile edge. We consider three offloading computing modes and focus on the coverage probability of each mode and corresponding ergodic rates. The resulting optimization problem is a mixed-integer and non-convex problem in the objective function and constraints. We propose a low-complexity suboptimal algorithm called Iteration of Convex Optimization and Nonlinear Programming (ICONP) to solve it. Numerical results verify the better performance of our proposed algorithm. Optimal computing ratios and APs’ density allocation contribute to the charge saving.

A Team Allocation Decision for Aircraft Fleet Maintenance

This paper is about a computer application for planning maintenance of aircraft fleet in what regards allocation of skilled technicians. The formalization for this planning is a mathematical programming problem written as a minimizing one. The decision variables are the allocation of the skilled technicians. The data are the number of available technicians, the working hours needed to accomplish maintenance, the costs due to the daily operation of facilities, and due to the fleet downtime. Although, the formalization is of a non-linear integer-programming problem, a transformation of variables from positive integer ones to Boolean ones is conceivable and allows a reformulation as a pure integer linear programming problem. Also, a relaxation of the integer variables to continuous ones allows for a relaxed formulation as a convex non-linear programming problem. A case study of two fleets illustrates both formulations: solved by the pyomo library calling the GLPK for the pure linear integer programming problem and calling solvers in Scipy library for the relaxed formulation. An inference presented about the results of the case study addresses the issue of using nonlinear programming to emulate the solution of linear integer programming.

A decentralized algorithm to combine topology control with network coding

Network coding and topology control techniques have been widely used to increase throughput and improve the lifetime of Wireless Sensor Networks (WSNs). This paper considers the simultaneous utilization of these techniques in a WSN and proposes convex non-linear programming. Since solving the problem for a large-scale and dynamic WSN is impractical and almost impossible, Lagrangian, sub-gradient and the decomposition methods are employed to provide a decentralized algorithm. In the proposed algorithm, a node makes the computations by acquiring local knowledge and information from its neighbors. This paper provides a mathematical language to build an analytic foundation for the design of a modularized and decentralized algorithm that provides transmission ranges and routes for a WSN. The simulation results show that increasing the number of sources, sinks, sensors, and traffic load leads to improving the lifetime which is acquired by the proposed algorithm.

PREDICTIVE CONTROL METHODS IN TASKS OF SEARCHING SADDLE POINTS

The article presents new methods for searching critical points of a function of several variables, including saddle points. Such problems are found in various fields of theoretical and practical science, for example, saddle-point construction lens design, machine and deep learning, problems of convex optimization and nonlinear programming (necessary and sufficient conditions for the solution are formulated using saddle points of the Lagrange function and proved in the Kuhn-Tucker theorem. When training neural networks, it is necessary to repeat the training process on large clusters and check the network's trainability at different loss functions and different network depth. Which means that thousands of new calculations are run, where each time the loss function is optimized on large amounts of data. So any acceleration in the process of finding critical points is a major advantage and saves computing resources. Many modern methods of searching saddle points are based on calculating the Hessian matrix, inverting this matrix, the scalar product of the gradient vector and the current vector, finding the full Lagrangian, etc. However, all these operations are computationally “expensive” and it would make sense to bypass such complex calculations. The idea of modifying the standard gradient methods used in the article is to apply fixed-point search schemes for nonlinear discrete dynamical systems for gradient descent problems. It is assumed that these fixed points correspond to unstable equilibrium positions, and there are large units among the multipliers of each equilibrium position. The averaged predictive control methods are used. Results of numerical modeling and visualization are presented in the form of two tables, which indicate basins of attraction for each critical point in each scheme, and statistical data by the convergence rates.

Inventory replenishment planning of a distribution system with storage capacity constraints and multi-channel order fulfilment

In this paper, an inventory replenishment planning problem in a two-echelon distribution system of Alibaba with a central distribution centre (CDC) and multiple front distribution centers (FDCs) is studied. In the system, the CDC replenishes its inventories of multiple products from external suppliers, while each FDC replenishes its inventories of the products from the CDC. These products are jointly replenished at each stock. This problem arises when Alibaba prepares its annual “double 11” promotion. Facing stochastic demands of the products in the promotion period, the objective of the problem is to optimally determine the inventory replenishment quantities of the products for each stock before the promotion so that the expected total profit of the system is maximized, subject to the storage capacity constraint of each stock. During the promotion, customer orders may be fulfilled through multiple channels, that is, a customer order may be fulfilled by its local FDC, other FDCs, or the CDC. This stochastic optimization problem is formulated as a convex nonlinear programming model and solved optimally by a piecewise linear approximation approach. The effectiveness of the model and the approach is demonstrated by numerical experiments on instances generated based on data of Alibaba. The results show that our model can lead to a significantly higher expected total profit compared with a deterministic planning model.

Real-Time Energy Management for Smart Homes

This article aims to develop a multiobjective optimization portfolio for real-time energy management in a smart home equipped with battery associated rooftop solar panels, lighting loads, air conditioners, and other smart home appliances. The energy management problem is framed for simultaneous minimization of the monetary energy cost and total dissatisfaction due to regulation in power consumption. The entire optimization portfolio is designed as a time average stochastic problem, which is simplified by the combination of queueing theory and Lyapunov optimization. The revised problem takes form of a mixed integer convex nonlinear programming, which is further solved using outer approximation approach. The proposed real-time home energy management framework needs only the current data regarding the random input parameters like renewable generation, energy price, and aggregated load demand, and does not call for their probabilistic estimation. Case study is carried out on a practical home data to proof efficacy of the proposed strategy and also the simulation outcomes are compared with one of the popular real-time energy management process named greedy algorithm.

The Linear and Asymptotically Superlinear Convergence Rates of the Augmented Lagrangian Method with a Practical Relative Error Criterion

In this paper, we conduct a convergence rate analysis of the augmented Lagrangian method with a practical relative error criterion designed in Eckstein and Silva [Mathematical Programming, 141, 319–348 (2013)] for convex nonlinear programming problems. We show that under a mild local error bound condition, this method admits locally a Q-linear rate of convergence. More importantly, we show that the modulus of the convergence rate is inversely proportional to the penalty parameter. That is, an asymptotically superlinear convergence is obtained if the penalty parameter used in the algorithm is increasing to infinity, or an arbitrarily Q-linear rate of convergence can be guaranteed if the penalty parameter is fixed but it is sufficiently large. Besides, as a byproduct, the convergence, as well as the convergence rate, of the distance from the primal sequence to the solution set of the problem is obtained.

An Optimal Multivariable Constrained Nonlinear (MVCNL) Stochastic Microgrid Planning and Operation Problem With Renewable Penetration

Microgrids (MGs) are promising archetypes for the penetration of renewable energy sources (RES) into the system for reducing carbon footprints. This article presents a cost-based optimization model using a stochastic programming to obtain the optimal size of distributed energy resources including RES. Uncertainties in the solar output and the load demand data are considered using multiple scenarios with random variations, which are based on beta distribution. The objective function is formulated as a cost-based multivariable constrained nonlinear convex programming problem. Other than planning costs, the objective function also includes per day running or operational costs such as unit commitment costs as well as the switching costs associated with the dispatchable units. The overall problem is simulated under MATLAB environment. Quantitative results indicate the impact of operational paradigm along with the planning of an MG in the form of cost analysis with different reliability conditions. An assessment of results in tabular form is presented, which incorporates the optimal sizing of MG under the different system reliability conditions and constraints. Some additional cases pertain to the generation and the load demand scenarios have also been included in the article.

A prediction-correction inexact alternating direction method for convex nonlinear second-order cone programming with linear constraints

The convex nonlinear second-order cone programming with linear constraints is equivalent to a separate structure convex programming. A prediction-correction inexact alternating direction method is proposed for the separate structure convex programming. In the proposed method, the convex objective function is not required to be Lipschitz continuous and only needs satisfy an inequality. The global convergence result is given. Numerical results demonstrate that our method is efficient for some random second-order cone programming problems in lower accuracy. In addition, our method can be extended to the convex nonlinear circular cone programming with linear constraints. We also give the simulation results of the three-fingered grasping force optimization problems.