Discretization algorithms for generalized semi-infinite programs with coupling equality constraints under local solution stability

Uncertainties in variables and parameters cause optimization problems to move away from globally-optimal and uncertain solutions. Practitioners resort to finding robust and reliable solutions in such situations. Bilevel optimization problems involving a hierarchy of two nested optimization problems have received a growing attention in the recent past due to their relevance in practice. While a number of studies on bilevel solution methodologies and applications are available for a deterministic setup, but studies on uncertainties in bilevel optimization are rare. In this paper, we suggest methodologies for handling uncertainty in both lower and upper level variables that may occur from different practicalities. For the first time, we perform a systematic study demonstrating the effect of uncertainties in each level along with the definition of robustness and reliability in the context of bilevel optimization. The issues and complexities introduced due to such uncertainties are then studied through a number of test cases, for brevity, we only show results on three test cases. Finally, two real-world bilevel problems involving uncertainties in their variables are solved. The study provides foundations and demon- strates viable directions for further research in uncertainty-based bilevel optimization problems.

We introduce a twice differentiable augmented Lagrangian for nonlinear optimization with general inequality constraints and show that a strict local minimizer of the original problem is an approximate strict local solution of the augmented Lagrangian. A novel augmented Lagrangian method of multipliers (ALM) is then presented. Our method is originated from a generalization of the Hestenes-Powell augmented Lagrangian, and is a combination of the augmented Lagrangian and the interior-point technique. It shares a similar algorithmic framework with existing ALMs for optimization with inequality constraints, but it can use the second derivatives and does not depend on projections on the set of inequality constraints. In each iteration, our method solves a twice continuously differentiable unconstrained optimization subproblem on primal variables. The dual iterates, penalty and smoothing parameters are updated adaptively. The global and local convergence are analyzed. Without assuming any constraint qualification, it is proved that the proposed method has strong global convergence. The method may converge to either a Karush-Kuhn-Tucker (KKT) point or a singular stationary point when the converging point is a minimizer. It may also converge to an infeasible stationary point of nonlinear program when the problem is infeasible. Furthermore, our method is capable of rapidly detecting the possible infeasibility of the solved problem. Under suitable conditions, it is locally linearly convergent to the KKT point, which is consistent with ALMs for optimization with equality constraints. The preliminary numerical experiments on some small benchmark test problems demonstrate our theoretical results.

This paper examines the effect of mating restriction on the search ability of EMO algorithms. First we propose a simple but flexible mating restriction scheme where a pair of similar (or dissimilar) individuals is selected as parents. In the proposed scheme, one parent is selected from the current population by the standard binary tournament selection. Candidates for a mate of the selected parent are winners of multiple standard binary tournaments. The selection of the mate among multiple candidates is based on the similarity (or dissimilarity) to the first parent. The strength of mating restriction is controlled by the number of candidates (i.e., the number of tournaments used for choosing candidates from the current population). Next we examine the effect of mating restriction on the search ability of EMO algorithms to find all Pareto-optimal solutions through computational experiments on small test problems using the SPEA and the NSGA-II. It is shown that the choice of dissimilar parents improves the search ability of the NSGA-II on small test problems. Then we further examine the effect of mating restriction using large test problems. It is shown that the choice of similar parents improves the search ability of the SPEA and the NSGA-II to efficiently find near Pareto-optimal solutions of large test problems. Empirical results reported in this paper suggest that the proposed mating restriction scheme can improve the performance of EMO algorithms for many test problems while its effect is problem-dependent and algorithm-dependent.

We examine bilevel mixed-integer programs whose constraints and objective functions depend on both upper- and lower-level variables. The class of problems we consider allows for nonlinear terms to appear in both the constraints and the objective functions, requires all upper-level variables to be integer, and allows a subset of the lower-level variables to be integer. This class of bilevel problems is difficult to solve because the upper-level feasible region is defined in part by optimality conditions governing the lower-level variables, which are difficult to characterize because of the nonconvexity of the follower problem. We propose an exact finite algorithm for these problems based on an optimal-value-function reformulation. We demonstrate how this algorithm can be tailored to accommodate either optimistic or pessimistic assumptions on the follower behavior. Computational experiments demonstrate that our approach outperforms a state-of-the-art algorithm for solving bilevel mixed-integer linear programs.

Existing distributed optimization algorithms often rely on the Lipschitz continuity of the objective function's gradient, but in practice, the Lipschitz constant is difficult to estimate, and the global Lipschitz continuity assumption may not hold. In this article, we propose two novel decentralized (proximal) algorithms, adaptive decentralized proximal primal--dual (ADPPD) and adaptive decentralized primal--dual (ADPD), which incorporate specially designed adaptive stepsizes within an improved primal-dual framework for composite optimization problems. These algorithms only require local estimates of cocoercivity and the Lipschitz modulus, eliminating the need for global Lipschitz continuity and avoiding the overly conservative stepsizes associated with a large Lipschitz constant. Moreover, the adaptive stepsizes are independent of the network, making the algorithms highly scalable. We provide detailed theoretical analyses to prove that ADPPD shows an ergodic convergence rate $\mathcal {O}({\scriptstyle \text {}^{\scriptstyle 1}}\hspace {-0.224em}/\hspace {-0.112em}{\scriptstyle k})$ when the smooth term $f_{i}$ and the nonsmooth term $g_{i}$ are convex, and ADPD shows a linear convergence rate when $f_{i}$ is strongly convex. Numerical experiments on least-squares and logistic regression problems confirm our theoretical results, demonstrating that our algorithms achieve faster convergence due to the better utilization of local Lipschitz continuity.

Solving the competitive discretionary service facility location problem

A fuzzy multi-objective optimization model for sustainable reverse logistics network design

In FY19 efforts were made on updating and verifying the PROTEUS suite in terms of depletion and fuel cycle capabilities, updated MSR capabilities, improving code usability, and supporting simulation of micro reactors. The fuel reloading and shuffling schemes as well as the built-in depletion solver were implemented to add the fuel cycle capability to PROTEUS-NODAL. PERSENT can run with PROTEUS-NODAL which now produce the CCCC interface files. The MSR simulation capability was further updated by making PROTEUS-NODAL coupled with SAM via the MOOSE wrapper. The high fidelity solvers of PROTEUS were updated to support the multiphysics simulation of SFRs and micro reactors. The coupled system of PROTEUS / FLUENT / ANLHTP was developed using Python-based external drivers, demonstrating the coupled simulation for small 3D test problems with heat pipe failure transients. In addition, the gamma transport capability was implemented to compute the detailed gamma distributions that determines the primary heat sources for the fuel assembly structure and non-fueled assemblies. Verification tests using fuel assembly and 2D core problems indicated very good agreement in heating distributions between PROTEUS and VARIANT / GAMSOR solutions. For user support, the NEAMS Workbench was updated to include PROTEUS-NODAL and further extended to handle PROTEUS-MOC for which the support of Workbench is limited, instead requiring the off-line generation of mesh files and corresponding cross section sets. The utility tools of PROTEUS were also updated to support users to model and simulate complicated and non-conventional geometry cores.

The subject of article is the kinematic and dynamic analysis of the parallel-sequential structure spatial mechanisms. The synthesis of a parallel-sequential structure mechanism for extended objects processing with kinematic chains made according to the l-coordinate scheme was carried out. Its analysis has been carried out. Kinematic and dynamic models of this mechanism have been developed. Mathematical models have been developed to study the kinematics and dynamics of a mechanism with an extended working area. The solution of the inverse problem on positions for this mechanism is given. This solution uses the classic coordinate transformation method. The final system of the constraints equations is obtained. The description of the methodology for solving the problem of velocities, which consists in differentiating the constraints equations (the Angeles-Gosselin method), is given. A system of implicit functions is presented. The solution of the problem of dynamics for a partial planar parallel structure mechanism with three kinematic chains is described. This solution was carried out using a control law that minimizes errors in speed, position, and acceleration. The mutual influence of drive motors on each other, as well as on the law of the executive link motion, is investigated. Full-scale and numerical experiments have been carried out on the developed models, and their results are presented. In a numerical experiment, a situation was simulated in which movement along one coordinate occurs according to a harmonic law, and no movement along the rest.KeywordsParallel-serial mechanismRobot with 5-DOFKinematic and dynamic analysisExtended working area

This paper proposed a new iterative method for solving nonlinear monotone equations with convex constraint and its applications in sparse signal reconstruction and image de-blurring problems. The method can be viewed as an improved adaptation of the generalized Hager-Zhang conjugate gradient method for unconstrained optimization. Unlike the latter which only converged globally for strongly convex functions when the Hager-Zhang parameter $ \theta_k $ lies in the interval $ \left(\frac{1}{4}, +\infty\right) $, the new method exhibited this attribute for nonlinear monotone and Lipschitz continuous functions without restriction for $ \theta_k $ under a more relaxed condition. The derivative-free structure of the method made it suitable for both smooth and non-smooth problems. Numerical experiments on benchmark test problems demonstrated the method's superior performance compared to some state-of-the-art algorithms. Furthermore, the algorithm was successfully applied to sparse signal recovery and image de-blurring problems in compressed sensing, confirming its practical effectiveness.

In the present work, we propose a two-phase fourth-order method for the approximate numerical solution for second-order non-linear two-point boundary value problems with Dirichlet boundary conditions. Our numerical approach is based on a finite difference and the solution of the problem at discrete points. Our method generates a system of equations, and the solution of the system of equations is considered an approximate solution to the problem. An essential analysis of the method is considered to ensure the performance of the method. A numerical experiment is carried out with model problems to test the performance in terms of efficiency and accuracy of the proposed method.

In this note we present a reconstructive algorithm for solving the cross-sectional pipe area from boundary measurements in a tree network with one inaccessible end. This is equivalent to reconstructing the first order perturbation to a wave equation on a quantum graph from boundary measurements at all network ends except one. The method presented here is based on a time reversal boundary control method originally presented by Sondhi and Gopinath for one dimensional problems and later by Oksanen to higher dimensional manifolds. The algorithm is local, so is applicable to complicated networks if we are interested only in a part isomorphic to a tree. Moreover the numerical implementation requires only one matrix inversion or least squares minimization per discretization point in the physical network. We present a theoretical solution existence proof, a step-by-step algorithm, and a numerical implementation applied to two numerical experiments.

In this paper, by considering distributed optimal control over a PDE, a gradient based iterative Algorithm is proposed for solving is proposed and analyzed. Galerkin finite element method is used for solving underlying PDE, then the adjoint base technique for derivative computation to implementation of the optimal control issue in preconditioned Newton's conjugate gradient method isused. The interface and connection between quadratic programming extracted from discretizing the problem and Newton's type method, as well as the convergence rate of the algorithm in each iteration is established. Updating control values at discretization points in each iteration yields optimal control of the problem, where the corresponding state values at these points approximate the desired function. Numerical experiments are presented for illustrating the theoretical results.

ADM FORMALISM APPLIED TO SOME SPACE-TIME MODELS USING EXCALC ALGEBRAIC PROGRAMMING

We consider resolvent splitting algorithms for finding a zero of the sum of finitely many maximally monotone operators. The standard approach to solving this type of problem involves reformulating as a two-operator problem in the product-space and applying the Douglas–Rachford algorithm. However, existing results for linear convergence cannot be applied in the product-space formulation due to a lack of appropriate Lipschitz continuity and strong monotonicity. In this work, we investigate a different approach that does not rely on the Douglas–Rachford algorithm or the product-space directly. We establish linear convergence of the ‘resolvent splitting with minimal lifting’ algorithm due to Malitsky and Tam for monotone inclusions with finitely many operators. Our results are then used to derive linear convergence of a primal–dual algorithm for convex minimization problems involving infimal convolutions. The theoretical results are demonstrated on numerical experiments in image denoising.