Second-order Cone Programming Solvers Research Articles

Adaptive XIGA shakedown analysis for problems with holes

Shakedown analysis plays a substantial role in safety assessment, especially in nuclear plant industry, chemical industry and civil engineering. This paper presents numerical investigations of shakedown problems with holes using the adaptive extended isogeometric analysis (XIGA). An adaptive strategy performed in the discretized framework of a kinematic approach and second-order cone programming (SOCP) is presented. The main idea is to integrate XIGA based on locally refined non-uniform rational B-splines (LR NURBS) into the SOCP to form an efficient approach for shakedown analysis. A SOCP form based on the von Mises yield criterion is established. By means of an effective SOCP solver, the arising optimization problem is then solved. In addition, the level set function is adopted to detect the position of holes. The L2-norm of plastic strain rates is used to guide local mesh refinement. The merits of the developed approach are its easy implementation and high computational efficiency. We consider several benchmark examples to illustrate the accuracy, reliability and efficiency of the developed method.

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

Quantum algorithms for Second-Order Cone Programming and Support Vector Machines

We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time O~(nrζκδ2log⁡(1/ϵ)) where r is the rank and n the dimension of the SOCP, δ bounds the distance of intermediate solutions from the cone boundary, ζ is a parameter upper bounded by n, and κ is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a δ-approximate ϵ-optimal solution of the given problem.Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision ϵ. We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O(nω+0.5) (here, ω is the matrix multiplication exponent, with a value of roughly 2.37 in theory, and up to 3 in practice). For the case of random SVM (support vector machine) instances of size O(n), the quantum algorithm scales as O(nk), where the exponent k is estimated to be 2.59 using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is 3.31 while that for a state-of-the-art SVM solver is 3.11.

Upper bound finite element limit analysis method with discontinuous quadratic displacement fields and remeshing in non-homogeneous clays

As is known, limit analysis is a common way to study stability and failure mechanism of geotechnical engineering. This paper presents an upper bound finite element limit analysis (UB-FELA) formulation for non-homogeneous clays using discontinuous quadratic displacement fields and second-order cone programming (SOCP). The details are provided about the application of SOCP to the UB-FELA formulation for non-homogeneous clays. To reduce the calculation error, the analytic expressions are proposed concerning the rate of external work done by body forces and power dissipation caused by the plastic deformation following discontinuous quadratic displacement fields. With SOCP solver MOSEK, large-scale optimization problems of UB-FELA can be achieved in minutes via a desktop computer. The method is then applied to analyzing the stability of an undrained square tunnel and an undrained plane strain heading. The numerical results are compared with those reported in related studies. This method helps to obtain highly precise upper bound solution and clear failure mechanism of the computational domain.

Joint optimization of location, inventory, and condition-based replacement decisions in service parts logistics

We model, analyze and study the effects of considering condition-based replacement of parts within an integrated Service Parts Logistics (SPL) system, where geographically dispersed customers’ products are serviced with new parts from network facilities. Conventional SPL models consider replacing the parts upon failure. This is true even for the latest models in which facility locations and their part stock levels are jointly optimized. Taking advantage of the increasingly affordable, continuous, and accurate collection of part condition data (via sensors and Internet-of-Things devices), we develop a new integrated model in which optimal conditions to replace the parts are decided along with facility locations and stock levels. We capture the part degradation, replacement and failure process using a Continuous Time Markov Chain (CTMC) and embed this into the integrated location and inventory model. The resulting formulation is a mixed-integer optimization model with quadratic constraints and is solved with a state-of-the-art second-order cone programming solver. Our extensive comparison with the traditional failure-based replacement model shows that optimizing replacement conditions in this integrated framework can provide significant cost savings (network, inventory, transportation and downtime costs) leading to different facility location, allocation and inventory decisions. We also study the effects of several important parameters on the condition-based replacement model, including facility costs, shipment speeds, replacement costs, part degradation parameters, and holding costs.

A linear programming based algorithm to solve a class of optimization problems with a multi-linear objective function and affine constraints

We present a linear programming based algorithm for a class of optimization problems with a multi-linear objective function and affine constraints. This class of optimization problems has only one objective function, but it can also be viewed as a class of multi-objective optimization problems by decomposing its objective function. The proposed algorithm exploits this idea and solves this class of optimization problems from the viewpoint of multi-objective optimization. The algorithm computes an optimal solution when the number of variables in the multi-linear objective function is two, and an approximate solution when the number of variables is greater than two. A computational study demonstrates that when available computing time is limited the algorithm significantly outperforms well-known convex programming solvers IPOPT and CVXOPT, in terms of both efficiency and solution quality. The optimization problems in this class can be reformulated as second-order cone programs, and, therefore, also be solved by second-order cone programming solvers. This is highly effective for small and medium size instances, but we demonstrate that for large size instances with two variables in the multi-linear objective function the proposed algorithm outperforms a (commercial) second-order cone programming solver.

Energy-efficient joint path-following for attitude-fixed space manipulators with bounds on completion time and motor voltage/current

This paper investigates the minimum-energy joint path-following problem for space manipulators whose base attitude is stabilized by reaction wheels. In the problem, manipulator joint path is specified for rest-to-rest motion and constraints are imposed as the upper bound on both motion completion time and the voltage/current limits of DC motors in manipulator joints and reaction wheels. We suggest a simple two-stage algorithm to address this problem. The algorithm first tries to find a global optimal solution by solving a relaxed convex problem. If the convex relaxation is not successful, then the algorithm solves subproblems iteratively to find a suboptimal solution. Since both problems are formulated as second-order cone programming (SOCP) form, they can be solved efficiently using dedicated SOCP solvers. The effectiveness of the proposed method is verified by numerical experiments.

Efficient numerical computations of yield stress fluid flows using second-order cone programming

This work addresses the numerical computation of the two-dimensional flow of yield stress fluids (with Bingham and Herschel–Bulkley models) based on a variational approach and a finite element discretization. The main goal of this paper is to propose an alternative optimization method to existing procedures such as penalization and augmented Lagrangian techniques. It is shown that the minimum principle for Bingham and Herschel–Bulkley yield stress fluid steady flows can, indeed, be formulated as a second-order cone programming (SOCP) problem, for which very efficient primal–dual interior point solvers are available. In particular, the formulation does not require any regularization of the visco-plastic model as is usually the case for existing techniques, avoiding therefore the difficult choice of the regularization parameter. Besides, it is also unnecessary to adopt a mixed stress–velocity approach or discretize explicitly auxiliary variables as frequently proposed in existing methods. Finally, the performance of dedicated SOCP solvers, like the Mosek software package, enables us to solve large-scale problems on a personal computer within seconds only. The proposed method will be validated on classical benchmark examples and used to simulate the flow generated around a plate during its withdrawal from a bath of yield stress fluid.

Novel Computational Implementations for Stability Analysis

By exploring the nature of the analogy between optimum trusses and optimum layouts of discontinuities, a novel numerical analysis method for rock/soil masses is proposed in this paper. The procedure is used to determine the critical layout of discontinuities and associated upper bound limit analysis for stability problems. The alternative approximation procedure to the traditional finite element method might involve discretization of a given body using a suitably large number of nodes laid out on a grid, with the failure mechanism comprising the most critical subset of potential discontinuities interconnecting these nodes. Potential discontinuities which interlink nodes laid out across the problem domain are permitted to crossover one another, giving a much wider search space than when such discontinuities are located only at the edges of finite elements of fixed topology. Highly efficient SOCP (second-order cone programming) solvers can be employed when certain popular failure criteria are specified (e.g. Hoek-Brown and Mohr-Coulomb). Stress/velocity singularities are automatically identified and visual interpretation of the output is straightforward. Several numerical examples including rock slope are studied by the new method, and the results are very close to those calculated by using analytical method and FEM.

Convex Optimization for Nonrigid Stereo Reconstruction

We present a method for recovering 3-D nonrigid structure from an image pair taken with a stereo rig. More specifically, we dedicate to recover shapes of nearly inextensible deformable surfaces. In our approach, we represent the surface as a 3-D triangulated mesh and formulate the reconstruction problem as an optimization problem consisting of data terms and shape terms. The data terms are model to image keypoint correspondences which can be formulated as second-order cone programming (SOCP) constraints using L(infinity) norm. The shape terms are designed to retaining original lengths of mesh edges which are typically nonconvex constraints. We will show that this optimization problem can be turned into a sequence of SOCP feasibility problems in which the nonconvex constraints are approximated as a set of convex constraints. Thanks to the efficient SOCP solver, the reconstruction problem can then be solved reliably and efficiently. As opposed to previous methods, ours neither involves smoothness constraints nor need an initial estimation, which enables us to recover shapes of surfaces with smooth, sharp and other complex deformations from a single image pair. The robustness and accuracy of our approach are evaluated quantitatively on synthetic data and qualitatively on real data.