Primal Solutions Research Articles

Chambolle–Pock’s Primal-Dual Method with Mismatched Adjoint

The primal-dual method of Chambolle and Pock is a widely used algorithm to solve various optimization problems written as convex-concave saddle point problems. Each update step involves the application of both the forward linear operator and its adjoint. However, in practical applications like computerized tomography, it is often computationally favourable to replace the adjoint operator by a computationally more efficient approximation. This leads to an adjoint mismatch in the algorithm. In this paper, we analyze the convergence of Chambolle–Pock’s primal-dual method under the presence of a mismatched adjoint in the strongly convex setting. We present an upper bound on the error of the primal solution and derive stepsizes and mild conditions under which convergence to a fixed point is still guaranteed. Furthermore we show linear convergence similar to the result of Chambolle–Pock’s primal-dual method without the adjoint mismatch. Moreover, we illustrate our results both for an academic and a real-world inspired application.

A novel decentralized approach to large-scale multi-agent MILPs

We address the optimal operation of a large-scale multi-agent system where agents have to set their own continuous and/or discrete decision variables so as to jointly minimize the sum of local linear performance indices while satisfying local and global linear constraints. When the number of discrete decision variables is large, solving the resulting Mixed Integer Linear Program becomes computationally demanding, and often impossible in practice. Inspired by some recent methods in the literature, we propose a decentralized iterative scheme that recovers computational tractability by decomposing the dual of the MILP problem into lower-dimensional MILPs, one per agent, and obtains feasibility of the recovered primal solution by introducing a fictitious tightening of the global constraints. The tightening is updated in an adaptive fashion according to an heuristic strategy which allows it to both increase and decrease throughout the iterations, depending on the mismatch between the recovered mixed-integer primal solution and the solution to the relaxed linear problem associated with the current tightening. The procedure is shown to be effective and to outperform state-of-the-art alternative resolution schemes in a benchmark example on optimal charging of a fleet of electric vehicles.

Self- Realization and Society in Gandhi’s Hind Swaraj (1909)

This paper tries to explore the role of self-realization as one of the primal solutions to psycho- socio-economic and political problems facing the individual, community, nation, and the world with reference to Mahatma Gandhi’s “Hind Swaraj”. Gandhi’s philosophy appears most clearly enunciated in Hind Swaraj (1909), a book written during his days in South Africa. The study investigates the issue of why the speaker gives more priority to self-realization while criticizing modern society as a major hindrance to free individuals, community, nation, and the world. The researcher argues that the speaker in this text wants to see his human world free from all kinds of socio-economic, political, and religious problems. The research approach adopted in this paper includes the opinions, and findings drawn from different research scholars and critiques. The conclusion drawn from this study is that self-realization should be seen as the main dimension of self-connectedness, authenticity, and the ability to realize and express ourselves through our work and thoughts.

A Function Approximation Approach for Parametric Optimization

We present a novel approach for approximating the primal and dual parameter-dependent solution functions of parametric optimization problems. We start with an equation reformulation of the first-order necessary optimality conditions. Then, we replace the primal and dual solutions with some approximating functions and find for some test parameters optimal coefficients as solution of a single nonlinear least-squares problem. Under mild assumptions it can be shown that stationary points are global minima and that the function approximations interpolate the solution functions at all test parameters. Further, we have a cheap function evaluation criterion to estimate the approximation error. Finally, we present some preliminary numerical results showing the viability of our approach.

Error transport equations implementation for discontinuous Galerkin methods

The procedure for performing higher-order discretization error estimation for Discontinuous Galerkin (DG) methods using Error Transport Equations (ETE) is presented in this work. The results obtained using the linearized ETE and the nonlinear ETE are compared for several steady state inviscid test cases containing completely smooth cases and cases with singularities or discontinuities. Prior work with the linearized version of ETE achieved improved results by using a relinearization process. This process is generalized to the nonlinear version of ETE and is more appropriately termed iterative correction when applied to both the linear and nonlinear versions. For smooth test cases, the discretization error estimates show expected order of accuracy which is higher than the order of accuracy of the primal solution. This holds for both the linearized and the nonlinear ETE. Discontinuities and singularities in the solution cause trouble in the discretization error estimation process. Discussion is made on the behavior of the ETE error estimates for non-smooth test cases to figure out potential solutions to the existing problems.

Duality of optimization problems with gauge functions

ABSTRACT Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimization problems. They presented a closed form of the dual formulation for the problem, and showed weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only weak duality but also strong duality. We also study necessary and sufficient optimality conditions associated to the problem. Moreover, we give sufficient conditions under which we can recover a primal solution from a Karush-Kuhn-Tucker point of the dual formulation. Finally, we discuss how to extend the above results to general optimization problems by considering the so-called perspective functions.

The hazardous orienteering problem

AbstractThis article studies the Hazardous Orienteering Problem (HOP), a variant of the more famous Orienteering Problem (OP). In the OP, a vehicle earns a profit for each customer it visits (e.g., to pick up a parcel) subject to an upper bound on the tour time. In the HOP, the parcels picked up at some customers have a probability of triggering a catastrophic event. The probability depends on how long the parcels travel on the vehicle. If any catastrophic event triggers, the entire collected profit is lost. The goal is to determine the tour that maximizes the expected profit. The problem has interesting applications in routing of hazardous material, cash‐in‐transit, and law enforcement. We propose a mixed‐integer nonlinear formulation and techniques both to obtain dual bounds and to produce primal solutions. Computational tests investigate the efficacy of the methods proposed and allow to gain insights into solution features.

On the Convergence of Overlapping Schwarz Decomposition for Nonlinear Optimal Control

We study the convergence properties of an overlapping Schwarz decomposition algorithm for solving nonlinear optimal control problems (OCPs). The algorithm decomposes the time domain into a set of overlapping subdomains, and solves all subproblems defined over subdomains in parallel. The convergence is attained by updating primal-dual information at the boundaries of overlapping subdomains. We show that the algorithm exhibits local linear convergence, and that the convergence rate improves exponentially with the overlap size. We also establish global convergence results for a general quadratic programming, which enables the application of the Schwarz scheme inside second-order optimization algorithms (e.g., sequential quadratic programming). The theoretical foundation of our convergence analysis is a sensitivity result of nonlinear OCPs, which we call “exponential decay of sensitivity” (EDS). Intuitively, EDS states that the impact of perturbations at domain boundaries (i.e., initial and terminal time) on the solution decays exponentially as one moves into the domain. Here, we expand a previous analysis available in the literature by showing that EDS holds for both primal and dual solutions of nonlinear OCPs, under uniform second-order sufficient condition, controllability condition, and boundedness condition. We conduct experiments with a quadrotor motion planning problem and a partial differential equations (PDE) control problem to validate our theory, and show that the approach is significantly more efficient than alternating direction method of multipliers and as efficient as the centralized interior-point solver.

Energy-Aware Cloud-Edge Collaborative Task Offloading with Adjustable Base Station Radii in Smart Cities

In smart cities, the computing power and battery life of terminal devices (TDs) can be effectively enhanced by offloading tasks to nearby base stations (BSs) with richer resources. With the goal of TDs being fully served and achieving low-carbon energy savings for the system, this paper investigates task offloading in cloud-edge collaborative heterogeneous scenarios with multiple BSs and TDs. According to the proportional relationship between the energy and coverage radii of BSs, a complete coverage task offloading model with adjustable BS radii is proposed. The task offloading problem is formulated as an integer linear program with multidimensional resource constraints to minimize the sum of energy consumption of BS coverage, offloading tasks to BSs and the cloud data center (CC). Since this task offloading problem is NP-hard, two approximate algorithms with polynomial time complexity are designed based on the greedy strategy of seeking the most energy-effective disk and the primal–dual method of constructing primal feasible solutions according to dual feasible solutions. Experimental results show that both the greedy and primal–dual algorithms can achieve good approximation performance, but each of them has its own advantages due to different design principles. The former is superior in execution time and energy consumption, while the latter has advantages in balancing loads among BSs and alleviating core network bandwidth pressure.

Maximum A Posteriori Inference of Random Dot Product Graphs via Conic Programming

We present a convex cone program to infer the latent probability matrix of a random dot product graph (RDPG). The optimization problem maximizes the Bernoulli maximum likelihood function with an added nuclear norm regularization term. The dual problem has a particularly nice form, related to the well-known semidefinite program relaxation of the maxcut problem. Using the primal-dual optimality conditions, we bound the entries and rank of the primal and dual solutions. Furthermore, we bound the optimal objective value and prove asymptotic consistency of the probability estimates of a slightly modified model under mild technical assumptions. Our experiments on synthetic RDPGs not only recover natural clusters, but also reveal the underlying low-dimensional geometry of the original data. We also demonstrate that the method recovers latent structure in the Karate Club Graph and synthetic U.S. Senate vote graphs and is scalable to graphs with up to a few hundred nodes.

Output-based error estimation and mesh adaptation for unsteady turbulent flow simulations

This paper presents a method for estimating output errors and adapting computational meshes in simulations of unsteady turbulent flows. The chaotic nature of such problems prevents a stable unsteady adjoint solution, and existing regularization techniques are costly for large simulations. The method presented foregoes the unsteady adjoint and instead relies on a field-inversion machine-learning (FIML) framework, which only requires unsteady primal solutions without full-state storage or checkpointing. The FIML model yields an adjoint for the averaged solution, which is combined with an averaged unsteady residual to obtain an output error estimate and adaptive indicator. This error estimate is shown to be accurate when the FIML model augments the original unsteady equations with corrections that are not excessively large. The unsteady residual comes from sampling fine-space residual evaluations during the unsteady simulation. A novel objective function based on an adjoint-weighted residual is presented for the field inversion to improve the ability of the FIML model to predict output errors and the domain-interior state. The localized output error drives adaptation of the mesh size and approximation order. Results for three aerodynamic problems ranging in Reynolds number demonstrate accuracy of the error estimates and efficiency of the computational meshes when compared to other adaptive strategies, including uniform and residual-based refinement.

Economic impacts of near-optimal solutions with non-convex pricing

We explore the relationship between consumer and producer surplus and optimality gaps in mixed-integer linear programming applied to electricity markets. We analyze a long-run adapted resource mix and introduce flexible demand. This allows for comparison of total producer and consumer surplus achieved across near-optimal solutions under different non-convex pricing models. Results indicate that for pricing models dependent on the primal solution, consumer surplus generally increases (although not monotonically) with increasing optimality and producer surplus correspondingly decreases. Short-run cost recovery is also possible over the operating horizon without make-whole payments for all models examined. These results are highly influenced by scarcity rent from price-setting elastic demand.

Attaining a good primal solution to the uncapacitated transportation problem

Transportation of products from sources to destinations with minimal total cost plays an important role in logistics and supply chain management. The Uncapacitated Transportation Problem (UTP) is a special case of network flow optimization problem. The prime objective of this UTP is to minimize the total cost of transporting products from origins to destinations subject to the respective supply and demand requirements. The UTP consists of special network structure. Due to the special structure of this problem, the transportation algorithm is preferred to solve it. The transportation algorithm consists of two major steps: 1) Finding an Initial Feasible Solution (IFS) to TP and 2) Examining the optimality of this IFS. A better IFS generates a lesser number of iterations to obtain a Minimal Total Cost Solution (MTCS). Recently, Juman and Nawarathne (2019)’s Method was introduced to find an IFS to UTP. In this paper, the Juman and Nawarathne (2019)’s Method is improved to get a better IFS to a UTP. A comparative study on a set of benchmark instances illustrates that the new improved method provides better primal solutions compared to the Juman and Nawarathne (2019)’s Method. The proposed method is found to yield the minimal total cost solutions to all the benchmark instances.

Exponential Convergence of Primal-Dual Dynamical System for Linear Constrained Optimization

Primal-dual dynamics (PDD) and its variants are prominent first-order continuous-time algorithms to determine the primal and dual solutions of a constrained optimization problem (COP). Due to the simple structure, they have received widespread attention in various fields, such as distributed optimization [1], power systems [2], and wireless communication [3]. In view of their wide applications, there are numerous theoretic studies on the convergence properties of PDD and its variants, including the exponential stability analysis [4]–[9].

Orlicz Space Regularization of Continuous Optimal Transport Problems

In this work we analyze regularized optimal transport problems in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, the aim is to find a transport plan, which is another Radon measure on the product of the sets, that has these two measures as marginals and minimizes the sum of a certain linear cost function and a regularization term. We focus on regularization terms where a Young’s function applied to the (density of the) transport plan is integrated against a product measure. This forces the transport plan to belong to a certain Orlicz space. The predual problem is derived and proofs for strong duality and existence of primal solutions of the regularized problem are presented. Existence of (pre-)dual solutions is shown for the special case of L^p-regularization for pge 2. Moreover, two results regarding varGamma -convergence are stated: The first is concerned with marginals that do not lie in the appropriate Orlicz space and guarantees varGamma -convergence to the original Kantorovich problem, when smoothing the marginals. The second result gives convergence of a regularized and discretized problem to the unregularized, continuous problem.

Monolithic One-Shot Optimization for Time-Periodic Flows Using Harmonic Balance

Adjoint methods for aerodynamic design optimization have been a considerable area of research in computational fluid dynamics. Several approaches have been developed to fully couple the flow, adjoint, and design equations for steady flows. However, only limited attempts have been made to apply those to unsteady flows using a monolithic approach. This study aims to fully couple the primal, adjoint, and design equations based on a harmonic balance technique allowing the unsteady flow optimization to be performed in a single monolithic computation. The harmonic balance is the appropriate method for such problems as it reformulates unsteady time-periodic problems as “mathematically steady,” where one does not need to store the entire time history of the primal solution. The novel unsteady technique will be applied to a time-periodic lid-driven cavity. Here, the Reynolds number is taken to be the design variable so as to match a target vorticity distribution at the first subtime level of the time-spectral (harmonic balance) solution. Finally, mean drag minimization of a rotationally oscillating cylinder in crossflow is performed. In all cases, it is shown that the proposed method is quite efficient, requiring less than a couple of hundred total iterations to identify the “optimum” conditions.

An accurate strategy for computing reaction forces and fluxes on trimmed locally refined meshes

Abstract The finite element method is classically based on nodal Lagrange basis functions defined on conforming meshes. In this context, total reaction forces are commonly computed from the so-called “nodal forces”, yielding higher accuracy and convergence rates than reactions obtained from the differentiated primal solution (“direct” method). The finite cell method and isogeometric analysis promise to improve the interoperability of computer-aided design and computer-aided engineering, enabling a direct approach to the numerical simulation of trimmed geometries. However, body-unfitted meshes preclude the use of classic nodal reaction algorithms. This work shows that the direct method can perform particularly poorly for immersed methods. Instead, conservative reactions can be obtained from equilibrium expressions given by the weak problem formulation, yielding superior accuracy and convergence rates typical of nodal reactions. This approach is also extended to non-interpolatory basis functions, such as the (truncated) hierarchical B-splines.

Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset’s elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation.

Composing Vessel Fleets for Maintenance at Offshore Wind Farms by Solving a Dual-Level Stochastic Programming Problem Using GRASP

Background: Dual-level stochastic programming is a technique that allows modelling uncertainty at two different levels, even when the time granularity differs vastly between the levels. In this paper we study the problem of determining the optimal fleet size and mix of vessels performing maintenance operations at offshore wind farms. In this problem the strategic planning spans decades, while operational planning is performed on a day-to-day basis. Since the operational planning level must somehow be taken into account when making strategic plans, and since uncertainty is present at both levels, dual-level stochastic programming is suitable. Methods: We present a heuristic solution method for the problem based on the greedy randomized adaptive search procedure (GRASP). To evaluate the operational costs of a given fleet, a novel fleet deployment heuristic (FDH) is embedded into the GRASP. Results: Computational experiments show that the FDH produces near optimal solutions to the operational day-to-day fleet deployment problem. Comparing the GRASP to exact methods, it produces near optimal solutions for small instances, while significantly improving the primal solutions for larger instances, where the exact methods do not converge. Conclusions: The proposed heuristic is suitable for solving realistic instances, and produces near optimal solution in less than 2 h.

Computational Methods for Scheduling the Charging and Assignment of an On-Site Shared Electric Vehicle Fleet

We investigate a fleet scheduling problem arising when a company has to manage its own fleet of electric vehicles. Aim is to assign given usage reservations to these vehicles and to devise a suitable charging plan for all vehicles while minimizing a cost function. We formulate the problem as a compact mixed integer linear program, which we strengthen in several ways. As this model is hard to solve in practice, we perform a Benders decomposition, which separates the problem into a master problem and a subproblem and solves them iteratively in an alternating manner. We perform the decomposition in two different ways. First we follow a more classical way, then we enrich the master problem making it stronger but also more complex and the subproblem smaller and simpler to solve. To improve the overall performance, we propose a problem-specific General Variable Neighborhood Search metaheuristic for solving the master problem in earlier iterations. Experimental results show that directly solving the complete mixed integer linear program usually performs well for small to some medium sized problem instances. For larger instances, however, it is not able to find any reasonable primal solutions anymore, while the Benders decomposition scales much better. Especially the variant with the heuristic delivers high quality solutions in reasonable time. The Benders decomposition with the more complex master problem also yields reasonable dual bounds and thus practically relevant quality guarantees for the larger instances.