Primal Problem Research Articles

A limiting analysis on regularization of singular SDP and its implication to infeasible interior-point algorithms

We consider primal-dual pairs of semidefinite programs and assume that they are singular, i.e., both primal and dual are either weakly feasible or weakly infeasible. Under such circumstances, strong duality may break down and the primal and dual might have a nonzero duality gap. Nevertheless, there are arbitrary small perturbations to the problem data which would make them strongly feasible thus zeroing the duality gap. In this paper, we conduct an asymptotic analysis of the optimal value as the perturbation for regularization is driven to zero. Specifically, we fix two positive definite matrices, I_p and I_d, say, (typically the identity matrices), and regularize the primal and dual problems by shifting their associated affine space by eta I_p and varepsilon I_d, respectively, to recover interior feasibility of both problems, where varepsilon and eta are positive numbers. Then we analyze the behavior of the optimal value of the regularized problem when the perturbation is reduced to zero keeping the ratio between eta and varepsilon constant. A key feature of our analysis is that no further assumptions such as compactness or constraint qualifications are ever made. It will be shown that the optimal value of the perturbed problem converges to a value between the primal and dual optimal values of the original problems. Furthermore, the limiting optimal value changes “monotonically” from the primal optimal value to the dual optimal value as a function of theta , if we parametrize (varepsilon , eta ) as (varepsilon , eta )=t(cos theta , sin theta ) and let trightarrow 0. Finally, the analysis leads us to the relatively surprising consequence that some representative infeasible interior-point algorithms for SDP generate sequences converging to a number between the primal and dual optimal values, even in the presence of a nonzero duality gap. Though this result is more of theoretical interest at this point, it might be of some value in the development of infeasible interior-point algorithms that can handle singular problems.

The Nonlinear Model of Intersectoral Linkages of Kazakhstan for Macroeconomic Decision-Making Processes in Sustainable Supply Chain Management

We provide a nonlinear model of intersectoral balance with constant elasticity of substitution (CES) production functions of industries and CES utility function of final consumer for the study of intersectoral linkages in the Kazakhstan economy. The model is formalized in terms of the primal problem of resource allocation and the corresponding Fenchel dual problem which solution gives costs of inputs of industries in a supply network. We identify the model with the actual data of the Input-Output tables of Kazakhstan and estimate the elasticity of substitution of production factors for the aggregated industry complexes. With the help of developed framework, we evaluate the inter-industry financial flows in the aggregated supply network for the period 2013–2020 and compare the results with the actual data of Kazakhstan. The developed framework can be used to support decision-making processes in sustainable supply chain management in a situation of the government economic policy change and external shocks. Using the developed framework, we evaluate the risks for Kazakhstan’s supply chains in scenario of sharp weakening of the national currency.

An Exact Algorithm for Multi-Task Large-Scale Inter-Satellite Routing Problem with Time Windows and Capacity Constraints

In the context of a low-orbit mega constellation network, we consider the large-scale inter-satellite routing problem with time windows and capacity constraints (ISRPTWC) with the goal of minimizing the total consumption cost, including transmission, resource consumption, and other environmentally impacted costs. Initially, we develop an integer linear programming model for ISRPTWC. However, a difficult issue when solving ISRPTWC is how to deal with complex time window constraints and how to reduce congestion and meet transmission capacity. Along this line, we construct a three-dimensional time-space state network aiming to comprehensively enumerate the satellite network state at any moment in time and a task transmission route at any given time and further propose a time-discretized multi-commodity network flow model for the ISRPTWC. Then, we adopt a dynamic programming algorithm to solve the single-task ISRPTWC. By utilizing a Lagrangian relaxation algorithm, the primal multi-task routing problem is decomposed into a sequence of single-task routing subproblems, with Lagrangian multipliers for individual task route nodes and links being updated by a subgradient method. Notably, we devise a novel idea for constructing the upper bound of the ISRPTWC. Finally, a case study using illustrative and real-world mega constellation networks is performed to demonstrate the effectiveness of the proposed algorithm.

Conic Duality for Multi-Objective Robust Optimization Problem

Duality theory is important in finding solutions to optimization problems. For example, in linear programming problems, the primal and dual problem pairs are closely related, i.e., if the optimal solution of one problem is known, then the optimal solution for the other problem can be obtained easily. In order for an optimization problem to be solved through the dual, the first step is to formulate its dual problem and analyze its characteristics. In this paper, we construct the dual model of an uncertain linear multi-objective optimization problem as well as its weak and strong duality criteria via conic duality. The multi-objective form of the problem is solved using the utility function method. In addition, the uncertainty is handled using robust optimization with ellipsoidal and polyhedral uncertainty sets. The robust counterpart formulation for the two uncertainty sets belongs to the conic optimization problem class; therefore, the dual problem can be built through conic duality. The results of the analysis show that the dual model obtained meets the weak duality, while the criteria for strong duality are identified based on the strict feasibility, boundedness, and solvability of the primal and dual problems.

Contract-based scheme for computational resource allocation in cloud-assisted parked vehicular edge computing

Having shown promising performance with high flexibility and efficiency in vehicular edge computing (VEC) network, the parked vehicles (PVs) received an increasing number of attentions in recent years. However, PVs’ residual battery power restricts their running time. In addition, there is still no alternate resource pool for the PVs to cope with the emergencies in the previous VEC framework. To alleviate these problems, we model a cloud-assisted parked vehicular edge computing (PVEC) framework, in which the PVs are classified based on their residual battery power. PVs corporate with the cloud servers (CSs) for the computational resources provision. In addition, we formulate the utilities of the service provider (SP) and PVs and design a contract-based resource allocation problem for the maximization of the SP’s utility. Considering that it is intractable to solve the optimization problem directly, the primal problem is simplified and decoupled into two sub-problems. To design the optimal contracts, we solve the sub-problems by Lagrangian multiplier method and dual function. Simulation results prove that the utilities of PVs can reach to the maximum when they choose the contract corresponding to their types. In addition, the simulation results illustrate the superiority of proposed scheme over previous schemes in improving the utilities of the SP and social welfare.

Inverse free reduced universum twin support vector machine for imbalanced data classification

Imbalanced datasets are prominent in real-world problems. In such problems, the data samples in one class are significantly higher than in the other classes, even though the other classes might be more important. The standard classification algorithms may classify all the data into the majority class, and this is a significant drawback of most standard learning algorithms, so imbalanced datasets need to be handled carefully. One of the traditional algorithms, twin support vector machines (TSVM), performed well on balanced data classification but poorly on imbalanced datasets classification. In order to improve the TSVM algorithm’s classification ability for imbalanced datasets, recently, driven by the universum twin support vector machine (UTSVM), a reduced universum twin support vector machine for class imbalance learning (RUTSVM) was proposed. The dual problem and finding classifiers involve matrix inverse computation, which is one of RUTSVM’s key drawbacks. In this paper, we improve the RUTSVM and propose an improved reduced universum twin support vector machine for class imbalance learning (IRUTSVM). We offer alternative Lagrangian functions to tackle the primal problems of RUTSVM in the suggested IRUTSVM approach by inserting one of the terms in the objective function into the constraints. As a result, we obtain new dual formulation for each optimization problem so that we need not compute inverse matrices neither in the training process nor in finding the classifiers. Moreover, the smaller size of the rectangular kernel matrices is used to reduce the computational time. Extensive testing is carried out on a variety of synthetic and real-world imbalanced datasets, and the findings show that the IRUTSVM algorithm outperforms the TSVM, UTSVM, and RUTSVM algorithms in terms of generalization performance.

Optimization of first-order Nicoletti boundary value problem with discrete and differential inclusions and duality

This paper is devoted to optimal control of the first-order Nicoletti boundary value problem (BVP) with discrete and differential inclusions (DFIs) and duality. First, we define Nicoletti-type problem with discrete inclusions, formulate optimality conditions for it and, based on the concept of infimal convolution, dual problems. Then, using the auxiliary discrete-approximate problem, we construct dual problems for Nicoletti DFIs and prove the duality theorems. Here, for the transition to the continuous problem, some results on the equivalence of locally adjoint mappings and support functions to the graph of set-valued mapping are proved. It turns out that the Euler–Lagrange type inclusions are ‘duality relations’ for both primal and dual problems, which means that a pair consisting of solutions to the primal and dual problems satisfies this extremal relation and vice versa. Finally, as an appication of the results obtained, we consider the first-order Nicoletti BVP with polyhedral DFIs.

Parallel-in-time adjoint-based optimization – application to unsteady incompressible flows

Gradient-based optimization algorithms, where gradient information is extracted using adjoint equations, are efficient but can quickly slow down when applied to unsteady and nonlinear flow problems. This is mainly due to the sequential nature of the algorithm, where the primal problem is first integrated forward in time, providing the initial condition for the adjoint problem, which is then integrated backward. In order to address the sequential nature of this optimization procedure parallel-in-time algorithms can be employed. However, the characteristics of the governing equations of interest in this work, and in particular, the divergence-free constraint (incompressibility effect) as well as the nonlinearity and the unsteadiness of the flow, make direct application of existing parallel-in-time algorithms less than straightforward. In this work, we introduce a parallel-in-time procedure, applied to the integration of the adjoint problem, which addresses all the existing constraints and allows quick access to local gradients. The performance of the proposed algorithm is assessed for both steady and unsteady actuation; in both cases it readily outperforms the sequential algorithm.

Topological optimization of a symmetrical adhesive joint. Island model of genetic algorithm

Modern additive technologies make it possible to create structures of variable thickness and of any shape. Thus, designers face problems of optimal design of a new type, and these are problems of topological optimization. Such problems are to determine the optimal form of the structure or the optimal distribution of material over the structure. As a rule, the criterion of optimality is the mass of the structure. However, the structure must retain its bearing capacity under a certain load. The symmetric two-shear adhesive joint of the main plate with two overlays of the same shape on both sides is the object of study in this article. The main goal of this study was to determine the optimal form of overlays with variable thicknesses under certain restrictions. The main restriction is the strength of the structure. Furthermore, additional restrictions are imposed on the minimum and maximum thickness of the overlay. Therefore, the solution to the problem is presented in the form of a set of the following tasks: building a mathematical model of the adhesive joint, building a numerical solution to the primal problem using the finite difference method, and building a genetic optimization algorithm. In the presented article, to improve the convergence of the genetic algorithm is proposed to use an island model that consists of several populations. The main feature of the proposed model of the genetic algorithm lies in the fact that on one of the "islands" mutations occur more frequently and with higher dispersion than on the other two "islands". On the one hand, this decision ensures a high rate of evolutionary selection, and on the other hand, the stability of the results is achieved. Several modeling problems are solved in this article. The main results of this research include the following: nonlinear dependence of the overlay length on the applied load was determined; restrictions on the minimum thickness of the overlay, which cause the appearance of a certain “plateau” at the edge of the overlay, the thickness of which is equal to the minimum allowable were defined.

A scheme for the analysis of primal stationary boundary value problems based on FE/FD multi-method

We present a non-iterative scheme for solving general linear stationary primal boundary value problems with an arbitrary number of degrees of freedom. The main goal is to combine the finite element method performance for domains with arbitrary geometry with the finite difference method efficiency related to its low computational cost. We apply a domain decomposition scheme to define rectangular subdomains (2D case) or rectangular hexahedral subdomains (3D case). A finite element mesh is generated in such a way that the elements in the regular regions are squares (2D case) or cubes (3D case). A suitable overlap scheme allows the finite element method to be used in regions of non-regular elements, while the finite difference method can be used in regions with regular elements (squares for 2D case or cubes for 3D case). We present six computational simulations showing the best performance in the CPU time of the proposed scheme compared to the classic finite element method while maintaining solution accuracy. • Reduction in the computational cost of assembling the matrices. • Solution time of the system of equations is preserved. • Feasible for problems with arbitrary degrees of freedom and non-homogeneous media. • Able to deal with several regular finite difference subdomains.

Implementation in Java of Algorithms for Semidefinite Optimization

Abstract We discuss the possibility of solving the semidefinite optimization problem using interior-point algorithms. We present the primal and dual semidefinite programming problems, and then determine the interior-point condition and the optimality criteria. We analyze the central path system and the modification of this, using the method of algebraically equivalent transformation. We use the Nesterov-Todd scaling technique to obtain the proper search directions. We give a modified version of the Nesterov-Todd step interior-point algorithm based on the implementation point of view. We present some numerical results based on a code implemented in the Java programming language. We compare the results obtained for the identity map and the square root function within the framework of the algebraically equivalent transformation technique.

A new randomized primal-dual algorithm for convex optimization with fast last iterate convergence rates

We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves and convergence rates in two cases: merely convexity and strong convexity, respectively, where k is the iteration counter and n is the number of block-coordinates. These rates are known to be optimal (up to a constant factor) when n = 1. Our convergence rates are obtained through three criteria: primal objective residual and primal feasibility violation, dual objective residual, and primal-dual expected gap. Moreover, our rates for the primal problem are on the last-iterate sequence. Our dual convergence guarantee requires additionally a Lipschitz continuity assumption. We specify our algorithm to handle two important special cases, where our rates are still applied. Finally, we verify our algorithm on two well-studied numerical examples and compare it with two existing methods. Our results show that the proposed method has encouraging performance on different experiments.

Integrating space syntax with spatial interaction

In this paper, we attempt to compare space syntax with spatial interaction. At one level, these two approaches to urban spatial structure are non-comparable. Space syntax is largely a descriptive technique for visualising spatial relations at the level of connections between places while spatial interaction is a predictive model that forecasts how much travel there will be between places. Space syntax articulates the system in terms of whether or not a physical link, usually at the level of the street, exists while spatial interaction predicts movements between all origins and destinations which are places often anchored in terms of the street network, but which at the level of prediction, assume connections between all places. Space syntax is grounded at a fine spatial scale while spatial interaction defines places as aggregates of activity in larger zones than the scale of the street system. The main output of space syntax is a connectivity matrix of step lengths between streets whereas in spatial interaction, such networks are predetermined, measurable in terms of Euclidean distance or generalised cost of travel, and the output is the volume of travel prior to this being assigned usually to a street network.There is however a fundamental way of relating the implicit network graph of spatial interaction to the explicit planar graph of the street network. We begin by assuming the planar graph of the network is conceived of as a primal problem of spatial interaction while the dual graph linking streets in the planar graph is the graph which is used in space syntax. We exploit this duality and show how we can move easily between spatial interaction as the primal and space syntax as the dual. This is rooted in a more fundamental graph – the bipartite graph which is a list of streets/arcs and their intersections/nodes from which the primal and dual emerge naturally. We explore various accessibility measures and show how they relate and correlate. We then go one step further and consider how various processes of random walking take place in these networks examining the steady states of the primal and dual problems in terms of the likelihood of a random walker visiting any node or street. We thus define primal and dual Markov chains that enable us to generate these probabilities. This provides a basic framework for comparing primal and dual in comparing spatial interaction with space syntax. We illustrate these measures on simple and easy to articulate graphs, extending this to a synthetic network of nearest neighbour links in Greater London based on 699 nodes and 1972 symmetric ‘streets’ between zones. This is a preliminary attack on the problem of linking these two approaches although many challenges remain.

High-Efficiency Flux Weakening Drive for IPMSM Based on Model Predictive Control

This article proposes a high-efficiency flux weakening (FW) drive based on the model predictive control (HEFW-MPC) for the interior permanent magnet synchronous motor (IPMSM). It is of great significance to improve the IPMSM’s efficiency due to its wide application and the urgent need for emission reduction. The FW drive can extend the speed range of IPMSM and reduce the gear ratio. The lower gear ratio is helpful to improve the mechanical transfer efficiency. However, the IPMSM’s loss has often been ignored. Therefore, the high-efficiency algorithm (HEA), concerning the copper and iron loss of IPMSM, is introduced to improve efficiency. Besides, the MPC is utilized as the controller to handle complex nonlinearities and constraints. However, it is difficult to directly solve the FW problem with multiple constraints, e.g., voltage limitation, current limitation, and HEA. Hence, the equivalent optimization is proposed to simplify the primal problem. The simulations and experiments verify that the presented HEFW-MPC is effective in driving IPMSM. What is more, the efficiency maps demonstrate that HEFW-MPC is helpful to improve the efficiency and extend the high-efficiency region of IPMSM.

Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields

Dual decomposition approaches in nonconvex optimization may suffer from a duality gap. This poses a challenge when applying them directly to nonconvex problems such as MAP-inference in a Markov random field (MRF) with continuous state spaces. To eliminate such gaps, this paper considers a reformulation of the original nonconvex task in the space of measures. This infinite-dimensional reformulation is then approximated by a semi-infinite one, which is obtained via a piecewise polynomial discretization in the dual. We provide a geometric intuition behind the primal problem induced by the dual discretization and draw connections to optimization over moment spaces. In contrast to existing discretizations which suffer from a grid bias, we show that a piecewise polynomial discretization better preserves the continuous nature of our problem. Invoking results from optimal transport theory and convex algebraic geometry we reduce the semi-infinite program to a finite one and provide a practical implementation based on semidefinite programming. We show, experimentally and in theory, that the approach successfully reduces the duality gap. To showcase the scalability of our approach, we apply it to the stereo matching problem between two images.

Huberized one-class support vector machine with truncated loss function in the primal space

One-class support vector machine (OCSVM) is an important tool in machine learning and has been extensively used for one-class classification problems. The traditional OCSVM solves the primal problem by solving the dual problem, which is a quadratic programming problem. However, the computation of the quadratic programming is cubic and the storage complexity is quadratic with problem scale, so it is inefficient for training large-scale problems. In this paper, we propose to train OCSVM in primal space directly. Unfortunately, owing to the non-differentiability of hinge loss used in OCSVM, the OCSVM cannot be solved by the gradient-based optimization method which is first-order method that converges fast. On the other hand, the hinge loss is unbounded which makes the OCSVM less robust to outliers. The outliers will make the decision boundary severely deviate from the optimal hyperplane. To overcome the drawbacks, a huberized truncated loss function which is a nonconvex differentiable function is proposed to improve the robustness of the OCSVM. The huberized truncated loss function is insensitive to outliers as a substitute for hinge loss in traditional OCSVM. In contrast to traditional OCSVM, the primal objective function of robust OCSVM is differentiable. Considering the non-convexity of the optimization problem, we employ an accelerated proximal gradient algorithm to solve the robust OCSVM in the primal space. The numerical experiments on benchmark datasets and handwritten digit datasets show that the proposed method not only improves the robustness of the OCSVM , but also can reduce the computational complexity.

Rolling horizon-based heuristics for solving a production-routing problem with price-dependent demand

This paper studies a production-routing problem with price-dependent demand (PRP-PD). The problem is formulated as a mixed-integer nonlinear program (MINLP) and solved within an outer approximation (OA) framework where the primal and master problems are solved iteratively. In order to solve the computationally expensive master problem, which is a mixed-integer program (MIP), four different rolling horizon-based heuristics are developed. The efficiency of the proposed heuristic algorithms is benchmarked against the existing methods in the literature to solve the PRP-PD for both linear and nonlinear demand functions. Our results show that by adopting proper freezing and simplification strategies, the rolling horizon-based heuristic can solve the large problem instances significantly faster than the existing methods in the literature, with little to no reduction in solution quality. For large problem instances with linear demand functions, the proposed heuristic provides solutions within 1.5–3 min. The quality of these solutions deviates at most 3.5% from the solutions provided by the existing exact methods, which take more than 2 h. Our solutions are also higher in quality compared to the near-optimal solutions provided in the literature, and they are achieved with a 90% reduction in computational time. With nonlinear demand functions, one-third of the instances require more than 2 h to be solved by the exact methods in the literature, while they are solved within 0.5–3.5 min using our heuristic, with less than 2% deviation in solution quality.

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

A new self-paced learning method for privilege-based positive and unlabeled learning

Positive and unlabeled learning (PU learning) is a kind of problem whose goal is learning a two-classes classifier with little proportion of positive samples and numerous unlabeled samples. A series of studies focus on how to extract most likely negative samples from the unlabeled samples, and then train a classifier with the labeled samples as supervised learning. Previous PU learning methods always ignore the additional information called privileged information, which is just provided during the training process while unavailable during testing. In this paper, we propose a novel self-paced algorithm for PU learning with privileged information (SPUPI). The proposed SPUPI extracts some reliable negative samples from unlabeled samples at first, and then generates weights for the unlabeled samples according to the similarity with each class. After that, it builds a more accurate classifier based on privileged information and similarity weights by self-paced learning. By taking the self-paced learning into training, we can build the model with a few labeled samples from easy to complex. We also solve the problem by transforming the primal problem of the proposed model into its dual problem and achieving the PU classifier. Various experiments on the practical datasets indicate that the SPUPI has a better performance compared with previous methods.

Optimized resource allocation and time partitioning for integrated communication, sensing, and edge computing network

The diverse computational tasks generated from advanced applications are becoming more difficult to process by mobile user equipments due to their limited computing capability and battery supply. With the fast development of wireless technology and infrastructure, edge computing is becoming a paradigm to alleviate these problems by offloading the computation tasks to the edge nodes with more computation resources. In addition, the integrated sensing and communication is a promising technology, where the wireless communication and radar sensing share unified hardware platform and radio resources. In this paper, the capabilities of communication, radar sensing and edge computing are integrated together in the proposed base station architecture to support the comprehensive services of data transmission, target sensing, and edge computing. Based on the proposed scheme, a resource allocation and time partitioning problem is investigated to jointly optimize time partitioning, computation task processing mode selection, spectrum resource allocation and target sensing location selection to maximize the weighted sum of task processing and communication performance while guaranteeing the radar sensing performance. Since the problem is non-convex, we decouple the primal problem into three subproblems which are solved separately. Simulation results show that our proposed scheme outperforms the typical relevant schemes and can converge within an acceptable iterations.