Interior-point Solvers Research Articles

Computational experience with a modified potential reduction algorithm for linear programming

We study the performance of a homogeneous and self-dual interior point solver for linear programming (LP) that is equipped with a continuously differentiable potential function. Our work is motivated by the apparent gap between the theoretical complexity results and long-step practical implementations in interior point algorithms. The potential function described here ensures a global linear polynomial-time convergence while providing the flexibility to integrate heuristics for generating the search directions and step length computations. Computational results on standard test problems show that LP problems are solved as efficiently (in terms of the number of iterations) as Mosek6 .

A locking-free stabilized kinematic EFG model for plane strain limit analysis

An element free Galerkin (EFG) based formulation for limit analysis of rigid-perfectly plastic plane strain problems is presented. In the paper it is demonstrated that volumetric locking and instability problems can be avoided by using a stabilized conforming nodal integration scheme. Furthermore, the stabilized EFG method described allows stable and accurate solutions to be obtained with minimal computational effort. The discrete kinematic formulation is cast in the form of a second-order cone problem, allowing efficient interior-point solvers to be used to obtain solutions. Finally, the performance of the stabilized EFG method is illustrated by considering several numerical examples.

A nonlinear programming approach for estimation of transmission parameters in childhood infectious disease using a continuous time model

Mathematical models can enhance our understanding of childhood infectious disease dynamics, but these models depend on appropriate parameter values that are often unknown and must be estimated from disease case data. In this paper, we develop a framework for efficient estimation of childhood infectious disease models with seasonal transmission parameters using continuous differential equations containing model and measurement noise. The problem is formulated using the simultaneous approach where all state variables are discretized, and the discretized differential equations are included as constraints, giving a large-scale algebraic nonlinear programming problem that is solved using a nonlinear primal-dual interior-point solver. The technique is demonstrated using measles case data from three different locations having different school holiday schedules, and our estimates of the seasonality of the transmission parameter show strong correlation to school term holidays. Our approach gives dramatic efficiency gains, showing a 40-400-fold reduction in solution time over other published methods. While our approach has an increased susceptibility to bias over techniques that integrate over the entire unknown state-space, a detailed simulation study shows no evidence of bias. Furthermore, the computational efficiency of our approach allows for investigation of a large model space compared with more computationally intensive approaches.

Global Convergence of Augmented Lagrangian Methods Applied to Optimization Problems with Degenerate Constraints, Including Problems with Complementarity Constraints

We consider global convergence properties of the augmented Lagrangian methods on problems with degenerate constraints, with a special emphasis on mathematical programs with complementarity constraints (MPCC). In the general case, we show convergence to stationary points of the problem under an error bound condition for the feasible set (which is weaker than constraint qualifications), assuming that the iterates have some modest features of approximate local minimizers of the augmented Lagrangian. For MPCC, we first argue that even weak forms of general constraint qualifications that are suitable for convergence of the augmented Lagrangian methods, such as the recently proposed relaxed positive linear dependence condition, should not be expected to hold and thus special analysis is needed. We next obtain a rather complete picture, showing that, under this context's usual MPCC-linear independence constraint qualification, feasible accumulation points of the iterates are guaranteed to be C-stationary for MPCC (better than weakly stationary), but in general need not be M-stationary (hence, neither strongly stationary). However, strong stationarity is guaranteed if the generated dual sequence is bounded, which we show to be the typical numerical behavior even though the multiplier set itself is unbounded. Experiments with the ALGENCAN augmented Lagrangian solver on the MacMPEC and DEGEN collections are reported, with comparisons to the SNOPT and filterSQP implementations of the SQP method, to the MINOS implementation of the linearly constrained Lagrangian method, and to the interior-point solvers IPOPT and KNITRO. We show that ALGENCAN is a very good option if one is primarily interested in robustness and quality of computed solutions.

An alternative use of the Riccati recursion for efficient optimization

In optimization routines used for on-line Model Predictive Control (MPC), linear systems of equations are solved in each iteration. This is true both for Active Set (AS) solvers as well as for Interior Point (IP) solvers, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of great interest to solve them efficiently. In high performance solvers for MPC, this is performed using Riccati recursions or generic sparsity exploiting algorithms. To be able to get this performance gain, the problem has to be formulated in a sparse way which introduces more variables. The alternative is to use a smaller formulation where the objective function Hessian is dense. In this work, it is shown that it is possible to exploit the structure also when using the dense formulation. More specifically, it is shown that it is possible to efficiently compute a standard Cholesky factorization for the dense formulation. This results in a computational complexity that grows quadratically in the prediction horizon length instead of cubically as for the generic Cholesky factorization.

Minimizing Condition Number via Convex Programming

In this paper we consider minimizing the spectral condition number of a positive semidefinite matrix over a nonempty closed convex set $\Omega$. We show that it can be solved as a convex programming problem, and moreover, the optimal value of the latter problem is achievable. As a consequence, when $\Omega$ is positive semidefinite representable, it can be cast into a semidefinite programming problem. We then propose a first-order method to solve the convex programming problem. The computational results show that our method is usually faster than the standard interior point solver SeDuMi [J. F. Sturm, Optim. Methods Softw., 11/12 (1999), pp. 625–653] while producing a comparable solution. We also study a closely related problem, that is, finding an optimal preconditioner for a positive definite matrix. This problem is not convex in general. We propose a convex relaxation for finding positive definite preconditioners. This relaxation turns out to be exact when finding optimal diagonal preconditioners.

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.

Limit analysis of plates and slabs using a meshless equilibrium formulation

AbstractA meshless Element‐Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs. In the formulation pure moment fields are approximated using a moving least‐squares technique, which means that the resulting fields are smooth over the entire problem domain. There is therefore no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable finite element formulation. The collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems. The combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes, and stable and accurate solutions can be obtained with minimal computational effort. The von Mises and Nielsen yield criteria which are used in the analysis of plates and slabs, respectively, are enforced by introducing second‐order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior‐point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems. Copyright © 2010 John Wiley & Sons, Ltd.

Explicit Sensor Network Localization using Semidefinite Representations and Facial Reductions

The sensor network localization (SNL) problem in embedding dimension r consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming (SDP) completion problem by using the linear mapping between Euclidean distance matrices (EDM) and semidefinite matrices. The resulting SDP is solved using primal-dual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances, we are able to efficiently solve many huge instances of this NP-hard problem to high accuracy by finding a representation of the minimal face of the SDP cone that contains the SDP matrix representation of the EDM. The main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the SDP cone that correspond to cliques of the SNL problem.

Interior-Point Method for Nuclear Norm Approximation with Application to System Identification

The nuclear norm (sum of singular values) of a matrix is often used in convex heuristics for rank minimization problems in control, signal processing, and statistics. Such heuristics can be viewed as extensions of $\ell_1$-norm minimization techniques for cardinality minimization and sparse signal estimation. In this paper we consider the problem of minimizing the nuclear norm of an affine matrix-valued function. This problem can be formulated as a semidefinite program, but the reformulation requires large auxiliary matrix variables, and is expensive to solve by general-purpose interior-point solvers. We show that problem structure in the semidefinite programming formulation can be exploited to develop more efficient implementations of interior-point methods. In the fast implementation, the cost per iteration is reduced to a quartic function of the problem dimensions and is comparable to the cost of solving the approximation problem in the Frobenius norm. In the second part of the paper, the nuclear norm approximation algorithm is applied to system identification. A variant of a simple subspace algorithm is presented in which low-rank matrix approximations are computed via nuclear norm minimization instead of the singular value decomposition. This has the important advantage of preserving linear matrix structure in the low-rank approximation. The method is shown to perform well on publicly available benchmark data.

An inexact interior-point method for system analysis

In this article, a primal-dual interior-point algorithm for semidefinite programming that can be used for analysing e.g. polytopic linear differential inclusions is tailored in order to be more computationally efficient. The key to the speedup is to allow for inexact search directions in the interior-point algorithm. These are obtained by aborting an iterative solver for computing the search directions prior to convergence. A convergence proof for the algorithm is given. Two different preconditioners for the iterative solver are proposed. The speedup is in many cases more than an order of magnitude. Moreover, the proposed algorithm can be used to analyse much larger problems as compared to what is possible with off-the-shelf interior-point solvers.

Upper and lower bound limit analysis of plates using FEM and second-order cone programming

This paper presents two novel numerical procedures to determine upper and lower bounds on the actual collapse load multiplier for plates in bending. The conforming Hsieh-Clough-Tocher (HCT) and enhanced Morley (EM) elements are used to discrete the problem fields. A Morley element with enhanced moment fields is used. The constant moment fields is added a quadratic mode in which the pressure is equilibrated by corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can be obtained. Once the displacement or moment fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization. In this paper, the optimization problems are formulated in the form of a standard second-order cone programming which can be solved using highly efficient interior point solvers. The procedures are tested by applying it to several benchmark plate problems and are found good agreement between the present upper and lower bound solutions and results in the literature.

Limit analysis decomposition and finite element mixed method

This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver, using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776–value to be compared to the best published lower bound 3.7752–by succeeding in solving a nonlinear optimization problem with millions of variables and constraints.

Reinforcement Learning Versus Model Predictive Control: A Comparison on a Power System Problem

This paper compares reinforcement learning (RL) with model predictive control (MPC) in a unified framework and reports experimental results of their application to the synthesis of a controller for a nonlinear and deterministic electrical power oscillations damping problem. Both families of methods are based on the formulation of the control problem as a discrete-time optimal control problem. The considered MPC approach exploits an analytical model of the system dynamics and cost function and computes open-loop policies by applying an interior-point solver to a minimization problem in which the system dynamics are represented by equality constraints. The considered RL approach infers in a model-free way closed-loop policies from a set of system trajectories and instantaneous cost values by solving a sequence of batch-mode supervised learning problems. The results obtained provide insight into the pros and cons of the two approaches and show that RL may certainly be competitive with MPC even in contexts where a good deterministic system model is available.

Limit analysis and convex programming: A decomposition approach of the kinematic mixed method

AbstractThis paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776—value to be compared with the best published lower bound 3.772—by succeeding in solving non‐linear optimization problems with millions of variables and constraints. Copyright © 2008 John Wiley & Sons, Ltd.

Preconditioning indefinite systems in interior point methods for large scale linear optimisation

We discuss the use of preconditioned conjugate gradients (CG) method for solving the reduced KKT systems arising in interior point algorithms for linear programming. The (indefinite) augmented system form of this linear system has a number of advantages, notably a higher degree of sparsity than the (positive definite) normal equations form. Therefore, we use the CG method to solve the augmented system and look for a suitable preconditioner. An explicit null space representation of linear constraints is constructed by using a nonsingular basis matrix identified from an estimate of the optimal partition in the linear program. This is achieved by means of recently developed efficient basis matrix factorisation techniques which exploit hyper-sparsity and are used in implementations of the revised simplex method. The approach has been implemented within the HOPDM interior point solver and applied to medium and large-scale problems from public domain test collections. Computational experience is encouraging.

Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with “cold start” on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performances of such warm-start strategies.

Identifying superfluous constraints within an interior-point algorithm for convex quadratic programming†

In this article, quadratic programming problems with strict convex objective functions f and linear constraints are considered. Based on a nonlinear separation Theorem, a complete characterization of constraints that are superfluous in an optimal point is given. It allows to derive sufficient conditions for deletion of restrictions. The corresponding conditions can easily be checked if upper bounds on the objective are available. The criteria are implemented within Mehrotra's primal-dual interior-point algorithm. Results with regards to how the identification of superfluous constraints works within that QP-solver are reported for a various number of randomly generated problem instances. Although the effect on the running time of the solution procedure was not yet examined, the article shows that the identification is highly efficient in early stages of of the algorithm. In such a way, the problem size could substantially be reduced. Such a reduction might potentially speed up interior-point solvers for large-scaled QP-problems. Dedicated to the memory of Prof. Dr. František Nožička

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of $${\mathcal{O}}(N^{6})$$ per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires $${\mathcal{O}}(N^3)$$ time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.

Joint Power Adaptation, Scheduling, and Routing for Ultra Wide Band Networks

A general cross-layer optimization problem, to maximize network efficiency (min-max power) of ad-hoc networks, is formulated including power adaptation, scheduling and routing functionalities. The non-convexity of the link capacity, high dimensionality of multi-hop routing and inter-layer interactions among the protocol layers, renders the problem hard. Conversion to an equivalent form, results in two clearly separable sub-problems, demonstrating the functionalities of the protocol layers. This decomposition allows the application of a simple shortest path based algorithm to the high-dimensional routing sub-problem. Further, in the case of UWB networks, the non-convex scheduling & power adaptation sub-problem can be effectively approximated and solved by applying a novel quadratic lower bound to the link capacity function. Using these algorithmic solutions to these sub-problems, an interior point solver generating solutions to the joint UWB network problem is developed. The various simulation results demonstrate interesting characteristics of the optimal routing and scheduling solutions, and provide benchmarks for UWB network design. Comparison with prior information theoretic capacity results, validates the importance of this cross-layer optimization framework