Relaxed Problem Research Articles

Relaxed optimal control problem for a finite horizon G-SDE with delay and its application in economics

This paper investigates the existence of a G-relaxed optimal control of a controlled stochastic differential delay equation driven by G-Brownian motion (G-SDDE in short). First, we show that optimal control of G-SDDE exists for the finite horizon case. We present as an application of our result an economic model, which is represented by a G-SDDE, where we studied the optimisation of this model. We connected the corresponding Hamilton–Jacobi–Bellman equation of our controlled system to a decoupled G-forward backward stochastic differential delay equation (G-FBSDDE in short). Finally, we simulate this G-FBSDDE to get the optimal strategy and cost.

A 0/1 Constrained Optimization Solving Sample Average Approximation for Chance Constrained Programming

Sample average approximation (SAA) is a tractable approach for dealing with chance constrained programming, a challenging stochastic optimization problem. The constraint of SAA is characterized by the 0/1 loss function, which results in considerable complexities in devising numerical algorithms. Most existing methods have been devised based on reformulations of SAA, such as binary integer programming or relaxed problems. However, the development of viable methods to directly tackle SAA remains elusive, let alone providing theoretical guarantees. In this paper, we investigate a general 0/1 constrained optimization, providing a new way to address SAA rather than its reformulations. Specifically, starting with deriving the Bouligand tangent and Fréchet normal cones of the 0/1 constraint, we establish several optimality conditions. One of them can be equivalently expressed by a system of equations, enabling the development of a semismooth Newton-type algorithm. The algorithm demonstrates a locally superlinear or quadratic convergence rate under standard assumptions along with nice numerical performance compared with several leading solvers. Funding: This research was supported by the National Key R&D Program of China [Grant 2023YFA1011100], the Fundamental Research Funds for the Central Universities, and the National Natural Science Foundation of China [Grant 12271309].

Order dispatch problem of the inter-city or inter-district ridesplitting service

This study formulates the order dispatch problem of the inter-city/inter-district ridesplitting service as an integer linear programming, and a solution framework that combines Lagrangian relaxation (LR) and alternating direction methods of multipliers is constructed. Specifically, LR problem is constructed by relaxing coupling constraints; based on this, the augmented Lagrangian relaxation (ALR) problem is constructed by adding a quadratic penalty term. Then, the ALR problem can be decomposed into a series of interdependent vehicle route subproblems by using linearisation technique and block coordinate descent method. The upper bound of the model is generated by ALR and a heuristic, and the solution quality can be evaluated by the lower bound provided by LR. Finally, numerous experiments are conducted to verify the effectiveness of proposed algorithms. The results show that the proposed algorithm could achieve an average cost saving of 10% and an average detour reduction of over 20% than the existing algorithm.

Online extrinsic parameters calibration of on-board stereo cameras based on certifiable optimization

The extrinsic parameters of on-board stereo cameras can be slightly altered due to temperature fluctuations, vibrations, and accidental impacts during automobile driving, leading to significant performance loss in dense stereo matching. In this paper, we propose an online calibration method based on certifiable optimization to address this issue. Initially, sparse feature points are collected based on plane distribution and disparity. A robust optimization model is then developed to minimize the epipolar error, utilizing iterative local optimization to eliminate outliers and determine the 5DOF extrinsic parameters. Subsequently, a relaxation problem is constructed using inliers, and global optimization is performed to certify that the locally optimal results are indeed globally optimal. Comparative experimental results demonstrate that the proposed method offers high accuracy and reliability. Additionally, the quality of the disparity map generated by our calibration method is comparable to that achieved through offline calibration.

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

A new perspective on denoising based on optimal transport

Abstract In the standard formulation of the classical denoising problem, one is given a probabilistic model relating a latent variable $\varTheta \in \varOmega \subset{\mathbb{R}}^{m} \; (m\ge 1)$ and an observation $Z \in{\mathbb{R}}^{d}$ according to $Z \mid \varTheta \sim p(\cdot \mid \varTheta )$ and $\varTheta \sim G^{*}$, and the goal is to construct a map to recover the latent variable from the observation. The posterior mean, a natural candidate for estimating $\varTheta $ from $Z$, attains the minimum Bayes risk (under the squared error loss) but at the expense of over-shrinking the $Z$, and in general may fail to capture the geometric features of the prior distribution $G^{*}$ (e.g. low dimensionality, discreteness, sparsity). To rectify these drawbacks, in this paper we take a new perspective on this denoising problem that is inspired by optimal transport (OT) theory and use it to study a different, OT-based, denoiser at the population level setting. We rigorously prove that, under general assumptions on the model, this OT-based denoiser is mathematically well-defined and unique, and is closely connected to the solution to a Monge OT problem. We then prove that, under appropriate identifiability assumptions on the model, the OT-based denoiser can be recovered solely from information of the marginal distribution of $Z$ and the posterior mean of the model, after solving a linear relaxation problem over a suitable space of couplings that is reminiscent of standard multimarginal OT problems. In particular, due to Tweedie’s formula, when the likelihood model $\{ p(\cdot \mid \theta ) \}_{\theta \in \varOmega }$ is an exponential family of distributions, the OT-based denoiser can be recovered solely from the marginal distribution of $Z$. In general, our family of OT-like relaxations is of interest in its own right and for the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.

Global optimization for large‐scale water network synthesis based on dynamic partition and adaptive bound tightening

AbstractThe synthesis of large‐scale integrated water networks is typically formulated as nonconvex mixed‐integer quadratic constrained programming (MIQCP) or QCP problems. With the complexity arising from bilinear terms in modeling mass flows of contaminants and binary variables representing the presence of units or streams, numerous local optima exist, thus presenting a significant optimization challenge. This study introduces a deterministic global optimization algorithm based on mixed‐integer programming (MIP) to tackle such problems. The approach involves dynamically strengthening the relaxed problems to converge towards the original problems. A simultaneous partition strategy is proposed combining locally uniform division with dynamic partitioned variables choosing. Furthermore, several adaptive bound contraction schemes are introduced to efficiently manage the size of the relaxed problems, assisting in accelerating the solution process. The algorithm's effectiveness and robustness are demonstrated with a large test set, showing superior performance compared to commercial solvers specifically on MIQCP problems.

Quasiconvex Functionals of (p, q)-Growth and the Partial Regularity of Relaxed Minimizers

We establish C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{C}^{\\infty }$$\\end{document}-partial regularity results for relaxed minimizers of strongly quasiconvex functionals F[u;Ω]:=∫ΩF(∇u)dx,u:Ω→RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathscr {F}[u;\\Omega ]:=\\int _{\\Omega }F(\ abla u)\ extrm{d}x,\\qquad u:\\Omega \\rightarrow \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}subject to a q-growth condition |F(z)|≦c(1+|z|q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|F(z)|\\leqq c(1+|z|^{q})$$\\end{document}, z∈RN×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\mathbb {R}^{N\ imes n}$$\\end{document}, and natural p-mean coercivity conditions on F∈C∞(RN×n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F\\in \ extrm{C}^{\\infty }(\\mathbb {R}^{N\ imes n})$$\\end{document} for the basically optimal exponent range 1≦p≦q<min{npn-1,p+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\leqq p\\leqq q<\\min \\{\\frac{np}{n-1},p+1\\}$$\\end{document}. With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from Schmidt’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}. We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca and Malý (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997).

Integrated aircraft routing and cargo routing problem for combination airlines

The combination airlines operate both passenger aircraft and freighter aircraft to meet passenger and cargo demand. At present, combination airlines employ a sequential approach to allocating their capacity for passenger and cargo demand. Nevertheless, implementing an integrated resource allocation procedure has the potential to improve overall resource allocation efficiency. In this paper, we introduce an integrated model to help combination airlines integrate their aircraft routing and cargo routing decisions to maximize the expected overall profits derived from both passenger and cargo demand. We considered the stochastic nature of passenger baggage and proposed a set of individual chance constraints to ensure the robustness of the integrated solution. We reformulate the chance constraints using piecewise linear approximation to ensure solution efficiency. In addition, we proposed a column-and-row generation based solution approach that removes the through-connection related constraints at the beginning of the solution process and then adds the columns and rows during the iterations as needed. We proved that the proposed column-and-row generation approach can obtain an optimal solution for the LP relaxation problem. The model and the solution approach were tested in a number of scenarios obtained from a major Chinese combination airline. The computational results show that the combination airline can improve their expected profits by integrating capacity allocation. The results also demonstrated that the proposed column-and-row generation solution approach can decrease the solution time of the integrated model. These findings indicate that the model and the solution method are useful and efficient tools for combination airlines when planning their aircraft and cargo routes.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Robust optimization algorithm for integrated crane assignment and scheduling in slab yard with uncertain arrival time

A slab yard served as a key component of the iron and steel plant. Effective slab yard crane assignment and scheduling directly affect the overall efficiency of the steel production process. This work addresses an uncertain slab yard crane assignment and scheduling problem (SYCAS) with the fluctuating arrival time. A stochastic programming model is established to obtain a robust solution that minimises the completion time of slab groups under uncertainty. Due to its NP-hardness, we developed an improved column and cut generation method (IC&CG), which divides the problem into a relaxed master problem (RMP) and a slave problem (SP). A new initialisation strategy for obtaining high-quality solutions and an acceleration strategy for discarding irrelevant scenarios are proposed. We use small-size instances to compare the performance of IC&CG with CPLEX and a generalised column and cut generation method in both deterministic and uncertain environments to validate the effectiveness of the method. For large-size instances, a lower bound (LB) of the optimal objective function is proposed to show the effectiveness of IC&CG. Furthermore, we prove the robustness of the method by sensitive analysis.

Phonon-Mediated Exciton Relaxation in Two-Dimensional Semiconductors: Selection Rules and Relaxation Pathways.

Exciton-phonon coupling (ExPC) is crucial for energy relaxation in semiconductors, yet the first-principles calculation of such coupling remains challenging, especially for two-dimensional (2D) systems. Here, an accurate method for calculating ExPC is developed and applied in exciton relaxation problems in monolayer WSe2. Considering the interplay between the exciton wave functions and electron-phonon coupling (EPC) matrix elements, we find that ExPC shows selection rules distinct from the ones of EPC. By employing the Wannier exciton model, we generalize these selection rules, which state that the angular quantum numbers of the exciton must match the winding numbers of the EPC matrix elements for the ExPC to be allowed. To verify our theory and method, we calculate intervalley exciton relaxation pathways, which agree well with a recent experiment.

A second-order particle Fokker-Planck model for rarefied gas flows

The direct simulation Monte Carlo (DSMC) method has become a powerful tool for studying rarefied gas flows. However, for the DSMC method to be effective, the cell size must be smaller than the mean free path, and the time step smaller than the mean collision time. These constraints make it difficult to use the DSMC method in multiscale rarefied gas flows. Over the past decade, the particle Fokker-Planck (FP) method has been studied to address computational cost issues in the near-continuum regime. To capture the main features of the Boltzmann equation, various FP models have been proposed, such as the quadratic entropic FP (Quad-EFP) and the ellipsoidal statistical FP (ESFP). Nevertheless, few studies have clearly demonstrated that the FP method offers a computational advantage over the DSMC method without sacrificing accuracy. This is because conventional particle FP methods have employed first-order accuracy schemes. The present study proposes a unified stochastic particle ESFP (USP-ESFP) model. This model improves the accuracy of shear stress and heat flux predictions. Additionally, a spatial interpolation scheme is introduced to the particle FP method. The numerical test cases include relaxation problem, Couette flows, Poiseuille flows, velocity perturbation, and hypersonic flows around a cylinder. The results show that the USP-ESFP model agrees well with both analytical and DSMC results. Furthermore, the USP-ESFP model is found to be less sensitive to cell size and time step than the DSMC method, resulting in a factor of four speed-up for the considered hypersonic flow around a cylinder.

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

Neural discovery of balance-aware polarized communities

Signed graphs are a model to depict friendly (positive) or antagonistic (negative) interactions (edges) among users (nodes). 2-Polarized-Communities (2pc) is a well-established combinatorial-optimization problem whose goal is to find two polarized communities from a signed graph, i.e., two subsets of nodes (disjoint, but not necessarily covering the entire node set) which exhibit a high number of both intra-community positive edges and negative inter-community edges. The state of the art in 2pc suffers from the limitations that (i) existing methods rely on a single (optimal) solution to a continuous relaxation of the problem in order to produce the ultimate discrete solution via rounding, and (ii) 2pc objective function comes with no control on size balance among communities. In this paper, we provide advances to the 2pc problem by addressing both these limitations, with a twofold contribution. First, we devise a novel neural approach that allows for soundly and elegantly explore a variety of suboptimal solutions to the relaxed 2pc problem, so as to pick the one that leads to the best discrete solution after rounding. Second, we introduce a generalization of 2pc objective function – termed γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-polarity – which fosters size balance among communities, and we incorporate it into the proposed machine-learning framework. Extensive experiments attest high accuracy of our approach, its superiority over the state of the art, and capability of function γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-polarity to discover high-quality size-balanced communities.

Fast Sparse Discriminative K-Means for Unsupervised Feature Selection.

Embedded feature selection approach guides subsequent projection matrix (selection matrix) learning through the acquisition of pseudolabel matrix to conduct feature selection tasks. Yet the continuous pseudolabel matrix learned from relaxed problem based on spectral analysis deviates from reality to some extent. To cope with this issue, we design an efficient feature selection framework inspired by classical least-squares regression (LSR) and discriminative K-means (DisK-means), which is called the fast sparse discriminative K-means (FSDK) for the feature selection method. First, the weighted pseudolabel matrix with discrete trait is introduced to avoid trivial solution from unsupervised LSR. On this condition, any constraint imposed into pseudolabel matrix and selection matrix is dispensable, which is significantly beneficial to simplify the combinational optimization problem. Second, the l2,p -norm regularizer is introduced to satisfy the row sparsity of selection matrix with flexible p . Consequently, the proposed FSDK model can be treated as a novel feature selection framework integrated from the DisK-means algorithm and l2,p -norm regularizer to optimize the sparse regression problem. Moreover, our model is linearly correlated with the number of samples, which is speedy to handle the large-scale data. Comprehensive tests on various data terminally illuminate the effectiveness and efficiency of FSDK.

ПРИНЦИП НАЛОЖЕНИЯ ДЕФОРМАЦИЙ В ТЕОРИИ ЖЕЛЕЗОБЕТОНА

The work substantiates the correctness of applying the principle of the overlay of deformations when deriving equations of the mechanical state of concrete that are significant in the theory of reinforced concrete in linear and nonlinear formulations. This principle consists in the summation at some point in time of partial deformations corresponding to partial increments of non-decreasing stress at successive previous points in time. Strains are represented by the sum of instantaneous strains and delayed creep strains at the moment of application. The superposition of partial creep strains is implemented according to the Boltzmann superposition principle known in the linear theory of creep - each partial increment of creep strain depends only on the magnitude and duration of the generating partial stress increment and does not depend on its other increments. When taking aging into account, postulated by this principle the mutual independence of partial increments of creep strain takes place relative to partial increments of the stress level. Representing the deformation as the sum of the initial elastic deformation and the creep deformation with a measure that takes into account the evolution of the elastic modulus leads to a different type of equation of state, which is perceived as incorrect in some publications. This perception of a different type of equation of state together with constructions, not related to the principle of the overlay of deformations, led these authors to declare that the principle of the overlay of deformations in the theory of creep is erroneous. In contrast to the traditional approach, a material (concrete, steel, wood, plastic) is considered as a combination of fractions (fibers, layers) with statistically distributed strengths. The concept of strength structure makes it possible to substantiate the principle of superimposing deformations in a nonlinear formulation. In this case, the structure of the nonlinear equation of state turns out to be identical to the structure of the linear equation. This circumstance is significant in applications, for example, when solving relaxation problems that are important for the long-term safety of building structures.

Direct Numerical Solutions to Stochastic Differential Equations with Multiplicative Noise.

Inspired by path integral solutions to the quantum relaxation problem, we develop a numerical method to solve classical stochastic differential equations with multiplicative noise that avoids averaging over trajectories. To test the method, we simulate the dynamics of a classical oscillator multiplicatively coupled to non-Markovian noise. When accelerated using tensor factorization techniques, it accurately estimates the transition into the bifurcation regime of the oscillator and outperforms trajectory-averaging simulations with a computational cost that is orders of magnitude lower.

Reduced Outer Space Algorithm for Globally Computing Affine Sum-of-Ratios Problems

This paper investigates the affine sum-of-ratios problem (SRP) which has numerous applications in many fields of economy and engineering. For globally computing the affine SRP, based on equivalent transformation and new affine relaxation problem, a reduced outer space branch-and-bound algorithm is designed. The designed algorithm has been proven to converge to a global optimal solution of the problem (SRP) eventually. Meanwhile, the maximum iteration time for the algorithm in the worst case is estimated by analyzing its computational complexity. Finally, numerical experimental results indicate that the presented algorithm is robust and effective.

Shaping Incentives through Measurement and Contracts

ABSTRACT I study productive activity, measurement, and compensation in a principal agent model that relaxes common restrictions on the action set of the agent, the distribution of performance measures, and the shape of the wage schedule. The solution to this relaxed problem unifies insights from extant theory and shares features with well-known empirical phenomena. In particular, the optimal outcome distribution has a kink, optimal measurement is conservative, and optimal wages ensure congruent incentives and resemble accounting-based bonus plans featuring a floor, hurdle bonus, incentive zone, and ceiling, with thresholds that may reference other performance measures. Beyond these specific insights, the paper provides a flexible framework for studying how incentives are shaped through measurement and contracts.