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593
Application of interiorpoint methods to model predictive control
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 1998
"... We present a structured interiorpoint method for the efficient solution of the optimal control problem in model predictive control (MPC). The cost of this approach is linear in the horizon length, compared with cubic growth for a naive approach. We use a discretetime Riccati recursion to solve the ..."
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Cited by 95 (6 self)
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We present a structured interiorpoint method for the efficient solution of the optimal control problem in model predictive control (MPC). The cost of this approach is linear in the horizon length, compared with cubic growth for a naive approach. We use a discretetime Riccati recursion to solve the linear equations efficiently at each iteration of the interiorpoint method, and show that this recursion is numerically stable. We demonstrate the effectiveness of the approach by applying it to three process control problems.
Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming
, 1997
"... Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by ..."
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Cited by 88 (15 self)
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Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primaldual interiorpoint algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.
Objectoriented software for quadratic programming
 ACM Transactions on Mathematical Software
, 2001
"... The objectoriented software package OOQP for solving convex quadratic programming problems (QP) is described. The primaldual interior point algorithms supplied by OOQP are implemented in a way that is largely independent of the problem structure. Users may exploit problem structure by supplying li ..."
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Cited by 87 (2 self)
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The objectoriented software package OOQP for solving convex quadratic programming problems (QP) is described. The primaldual interior point algorithms supplied by OOQP are implemented in a way that is largely independent of the problem structure. Users may exploit problem structure by supplying linear algebra, problem data, and variable classes that are customized to their particular applications. The OOQP distribution contains default implementations that solve several important QP problem types, including general sparse and dense QPs, boundconstrained QPs, and QPs arising from support vector machines and Huber regression. The implementations supplied with the OOQP distribution are based on such well known linear algebra packages as MA27/57, LAPACK, and PETSc. OOQP demonstrates the usefulness of objectoriented design in optimization software development, and establishes standards that can be followed in the design of software packages for other classes of optimization problems. A number of the classes in OOQP may also be reusable directly in other codes.
Optimal Design of a CMOS OpAmp via Geometric Programming
"... We describe a new method for determining component values and transistor dimensions for CMOS operational amplifiers (opamps). We observe that a wide variety of design objectives and constraints have a special form, i.e., theyareposynomial functions of the design variables. As a result the amplifi ..."
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Cited by 85 (9 self)
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We describe a new method for determining component values and transistor dimensions for CMOS operational amplifiers (opamps). We observe that a wide variety of design objectives and constraints have a special form, i.e., theyareposynomial functions of the design variables. As a result the amplifier design problem can be expressed as a special form of optimization problem called geometric programming, for which very efficient global optimization methods have been developed. As a consequence we can efficiently determine globally optimal amplifier designs, or globally optimal tradeoffs among competing performance measures such as power, openloop gain, and bandwidth. Our method therefore yields completely automated synthesis of (globally) optimal CMOS amplifiers, directly from specifications. In this paper we apply this method to a specific, widely used operational amplifier architecture, showing in detail how to formulate the design problem as a geometric program. We compute globally optimal tradeoff curves relating performance measures such as power dissipation, unitygain bandwidth, and openloop gain. We show how the method can be used to synthesize robust designs, i.e., designs guaranteed to meet the specifications for a variety of process conditions and parameters.
Multiplicative Updates for Nonnegative Quadratic Programming in Support Vector Machines
 in Advances in Neural Information Processing Systems 15
, 2002
"... We derive multiplicative updates for solving the nonnegative quadratic programming problem in support vector machines (SVMs). The updates have a simple closed form, and we prove that they converge monotonically to the solution of the maximum margin hyperplane. The updates optimize the traditiona ..."
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Cited by 81 (7 self)
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We derive multiplicative updates for solving the nonnegative quadratic programming problem in support vector machines (SVMs). The updates have a simple closed form, and we prove that they converge monotonically to the solution of the maximum margin hyperplane. The updates optimize the traditionally proposed objective function for SVMs. They do not involve any heuristics such as choosing a learning rate or deciding which variables to update at each iteration. They can be used to adjust all the quadratic programming variables in parallel with a guarantee of improvement at each iteration. We analyze the asymptotic convergence of the updates and show that the coefficients of nonsupport vectors decay geometrically to zero at a rate that depends on their margins. In practice, the updates converge very rapidly to good classifiers.
Preconditioning indefinite systems in interior point methods for optimization
 Computational Optimization and Applications
, 2004
"... Abstract. Every Newton step in an interiorpoint method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable il ..."
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Cited by 65 (17 self)
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Abstract. Every Newton step in an interiorpoint method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable illconditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used. Keywords: interiorpoint methods, iterative solvers, preconditioners 1.
Parallel InteriorPoint Solver for Structured Quadratic Programs: Application to Financial Planning Problems
, 2003
"... Many practical largescale optimization problems are not only sparse, but also display some form of blockstructure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is blockstructured itself. Problems with these charact ..."
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Cited by 62 (21 self)
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Many practical largescale optimization problems are not only sparse, but also display some form of blockstructure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is blockstructured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling. We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its objectoriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra. The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as
An interior point algorithm for largescale nonlinear . . .
, 2002
"... Nonlinear programming (NLP) has become an essential tool in process engineering, leading to prot gains through improved plant designs and better control strategies. The rapid advance in computer technology enables engineers to consider increasingly complex systems, where existing optimization codes ..."
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Cited by 61 (3 self)
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Nonlinear programming (NLP) has become an essential tool in process engineering, leading to prot gains through improved plant designs and better control strategies. The rapid advance in computer technology enables engineers to consider increasingly complex systems, where existing optimization codes reach their practical limits. The objective of this dissertation is the design, analysis, implementation, and evaluation of a new NLP algorithm that is able to overcome the current bottlenecks, particularly in the area of process engineering. The proposed algorithm follows an interior point approach, thereby avoiding the combinatorial complexity of identifying the active constraints. Emphasis is laid on exibility in the computation of search directions, which allows the tailoring of the method to individual applications and is mandatory for the solution of very large problems. In a fullspace version the method can be used as general purpose NLP solver, for example in modeling environments such as Ampl. The reduced space version, based on coordinate decomposition, makes it possible to tailor linear algebra
Interior point methods for massive support vector machines
, 2003
"... We investigate the use of interiorpoint methods for solving quadratic programming problems with a small number of linear constraints, where the quadratic term consists of a lowrank update to a positive semidefinite matrix. Several formulations of the support vector machine fit into this category ..."
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Cited by 56 (1 self)
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We investigate the use of interiorpoint methods for solving quadratic programming problems with a small number of linear constraints, where the quadratic term consists of a lowrank update to a positive semidefinite matrix. Several formulations of the support vector machine fit into this category. An interesting feature of these particular problems is the volume of data, which can lead to quadratic programs with between 10 and 100 million variables and, if written explicitly, a dense Q matrix. Our code is based on OOQP, an objectoriented interiorpoint code, with the linear algebra specialized for the support vector machine application. For the targeted massive problems, all of the data is stored out of core and we overlap computation and input/output to reduce overhead. Results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable. Key words. support vector machine, interiorpoint method, linear algebra AMS subject classifications.