Quadratic Semi-infinite Programming Research Articles

Learning Data Manifolds with a Cutting Plane Method.

We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation methods rely on sampling large numbers of training examples from these manifolds. Instead, we propose an iterative algorithm, [Formula: see text], based on a cutting plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of [Formula: see text] are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet data set. Our results indicate that [Formula: see text] is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin methods using data augmentation methods.

Convex approximations of analytic functions

We introduce a method for constructing the best approximation of an analytic function in a subclass K∗⊂K of convex functions, in the sense of the L2 norm. The construction is based on solving a certain semi-infinite quadratic programming problem, which may be of independent interest.

Monotone Smoothing Splines using General Linear Systems

In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second-order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval which has the cardinality of the continuum. To solve this problem, we first formulate the problem as a semi-infinite quadratic programming problem, and then we adopt a discretization technique to obtain a finite-dimensional quadratic programming problem. It is shown that the solution of the finite-dimensional problem always satisfies the infinite-dimensional monotonicity constraint. It is also proved that the approximated solution converges to the exact solution as the discretization grid-size tends to zero. An example is presented to show the effectiveness of the proposed method.

Design of spectrally efficient ultra-wideband waveforms using Hermite–Rodriguez functions

Spectral efficiency is a major requirement in ultra-wideband (UWB) communication systems because of very low power spectral density (PSD) regulations imposed on the transmitted signals. Besides, large number of orthogonal waveforms is highly desirable in multiuser systems. In this study, the design of a set of spectrally efficient orthogonal waveforms is investigated. The design of UWB waveforms based on orthogonal Hermite–Rodriguez basis functions and linear frequency modulation (LFM) is proposed herein. The design is formulated as a quadratic semi-infinite programming problem with spectral and orthogonality constraints. Numerical examples are presented to illustrate the effectiveness of Hermite–Rodriguez-based waveforms with the proposed optimisation approach. Comparison with a recent result shows that the proposed approach compares favourably with those reported in the literatures in terms of the number of waveforms can be produced and the achievable spectral efficiency of the designed waveforms.

Complex digital Laguerre filter design with weighted least square error subject to magnitude and phase constraints

Complex coefficient optimal digital Laguerre filter design with arbitrary asymmetric frequency response is developed. The optimization scheme is based on the weighted least square error criterion with complex Chebyshev error used as its magnitude and phase constraints. Developed version of the active set method has been exploited to solve the resultant semi-infinite quadratic programming problem. A quick and practical approach is proposed to evaluate the suboptimal value of the Laguerre parameter close to its optimum value. The design technique can be applied to design of asymmetric FIR filters just by setting the Laguerre parameter equal to zero. The accuracy of the processes has been verified by illustrating some numerical examples.

Numerical method for a class of optimal control problems subject to nonsmooth functional constraints

In this paper, we consider a class of optimal control problems which is governed by nonsmooth functional inequality constraints involving convolution. First, we transform it into an equivalent optimal control problem with smooth functional inequality constraints at the expense of doubling the dimension of the control variables. Then, using the Chebyshev polynomial approximation of the control variables, we obtain an semi-infinite quadratic programming problem. At last, we use the dual parametrization technique to solve the problem.

Design of Interpolative Sigma Delta Modulators Via Semi-Infinite Programming

This correspondence considers the optimized design of interpolative sigma delta modulators (SDMs). The first optimization problem is to determine the denominator coefficients. The objective of the optimization problem is to minimize the passband energy of the denominator of the loop filter transfer function (excluding the dc poles) subject to the continuous constraint of this function defined in the frequency domain. The second optimization problem is to determine the numerator coefficients in which the cost function is to minimize the stopband ripple energy of the loop filter subject to the stability condition of the noise transfer function (NTF) and signal transfer function (STF). These two optimization problems are actually quadratic semi-infinite programming (SIP) problems. By employing the dual-parameterization method, global optimal solutions that satisfy the corresponding continuous constraints are guaranteed if the filter length is long enough. The advantages of this formulation are the guarantee of the stability of the transfer functions, applicability to design of rational infinite-impulse-response (IIR) filters without imposing specific filter structures, and the avoidance of iterative design of numerator and denominator coefficients. Our simulation results show that this design yields a significant improvement in the signal-to-noise ratio (SNR) and have a larger stability range, compared with the existing designs

Optimal design of nonuniform FIR transmultiplexer using semi-infinite programming

This correspondence considers an optimum nonuniform finite impulse response (FIR) transmultiplexer design problem subject to specifications in the frequency domain. Our objective is to minimize the sum of the ripple energy for all the individual filters, subject to the specifications on amplitude and aliasing distortions, and to the passband and stopband specifications for the individual filters. This optimum nonuniform transmultiplexer design problem can be formulated as a quadratic semi-infinite programming problem. The dual parametrization algorithm is extended to this nonuniform transmultiplexer design problem. If the lengths of the filters are sufficiently long and the set of decimation integers is compatible, then a solution exists. Since the problem is formulated as a convex problem, if a solution exists, then the solution obtained is unique, and the local solution is a global minimum.

Design of nonuniform near allpass complementary FIR filters via a semi-infinite programming technique

In this correspondence, we consider the problem of designing a set of nonuniform near allpass complementary finite impulse response (FIR) filters. This problem can be formulated as a quadratic semi-infinite programming problem, where the objective is to minimize the sum of the ripple energy for the individual filters, subject to the passband and stopband specifications as well as to the allpass complementary specification. The dual parameterization method is used for solving the linear quadratic semi-infinite programming problem.

An Approximation Approach to Non-strictly Convex Quadratic Semi-infinite Programming

We present in this paper a numerical method for solving non-strictly-convex quadratic semi-infinite programming including linear semi-infinite programming. The proposed method transforms the problem into a series of strictly convex quadratic semi-infinite programming problems. Several convergence results and a numerical experiment are given.

A New Quadratic Semi-infinite Programming Algorithm Based on Dual Parametrization

The so called dual parameterization method for quadratic semi-infinite programming (SIP) problems is developed recently. A dual parameterization algorithm is also proposed for numerical solution of such problems. In this paper, we present and improved adaptive algorithm for quadratic SIP problems with positive definite objective and multiple linear infinite constraints. In each iteration of the new algorithm, only a quadratic programming problem with a limited dimension and a limited number of constraints is required to be solved. Furthermore, convergence result is given. The efficiency of the new algorithm is shown by solving a number of numerical examples.

A solution method for combined semi-infinite and semi-definite programming

AbstractIn this paper, we develop a discretisation algorithm with an adaptive scheme for solving a class of combined semi-infinite and semi-definite programming problems. We show that any sequence of points generated by the algorithm contains a convergent subsequence; and furthermore, each accumulation point is a local optimal solution of the combined semi-infinite and semi-definite programming problem. To illustrate the effectiveness of the algorithm, two specific classes of problems are solved. They are relaxations of quadratically constrained semi-infinite quadratic programming problems and semi-infinite eigenvalue problems.

An Adaptive Dual Parametrization Algorithm for Quadratic Semi-infinite Programming Problems

The so called dual parametrization method for quadratic semi-infinite programming (SIP) problems is developed recently for quadratic SIP problems with a single infinite constraint. A dual parametrization algorithm is also proposed for numerical solution of such problems. In this paper, we consider quadratic SIP problems with positive definite objective and multiple linear infinite constraints. All the infinite constraints are supposed to be continuously dependent on their index variable on a compact set which is defined by a number equality and inequalities. We prove that in the multiple infinite constraint case, the minimu parametrization number, just as in the single infinite constraint case, is less or equal to the dimension of the SIP problem. Furthermore, we propose an adaptive dual parametrization algorithm with convergence result. Compared with the previous dual parametrization algorithm, the adaptive algorithm solves subproblems with much smaller number of constraints. The efficiency of the new algorithm is shown by solving a number of numerical examples.

A unified quadratic semi-infinite programming approach to time and frequency domain constrained digital filter design

A unified quadratic semi-infinite programming approach is introduced to solve digital filter design problems with time or frequency-domain specification. An algorithm based on this approach is developed and the corresponding convergence result is presented. This computational method is then applied to the optimum filter design problems subject to time and frequency domain specifications, namely the time domain envelope constrained filter design problems and the frequency- domain least square FIR filter design problems. For illustration, two examples are given.

Solving quadratic semi-infinite programming problems by using relaxed cutting-plane scheme

This paper studies the cutting-plane approach for solving quadratic semi-infinite programming problems. Several relaxation techniques and their combinations are proposed and discussed. A flexible convergence proof is provided to cover different settings of a relaxation scheme. The implementation issues are addressed with some numerical experiments to illustrate the computational behavior of each different combination.

A semi-infinite quadratic programming algorithm with applications to array pattern synthesis

This paper presents a new extended active set strategy for optimizing antenna arrays by semi-infinite quadratic programming. The optimality criterion is either to maximize the directivity of the antenna or to minimize its sidelobe energy when subjected to a specified peak sidelobe level. Additional linear constraints are used to form the mainlobe. The design approach is applied to a numerical example that deals with the design of a narrow-band circular antenna array for the far field.

Semi-Infinite Programming Approach to Continuously-Constrained Linear-Quadratic Optimal Control Problems

Consider the class of linear-quadratic (LQ) optimal control problems with continuous linear state constraints, that is, constraints imposed on every instant of the time horizon. This class of problems is known to be difficult to solve numerically. In this paper, a computational method based on a semi-infinite programming approach is given. The LQ optimal control problem is formulated as a positive-quadratic infinite programming problem. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the decision variable, an approximate quadratic semi-infinite programming problem is obtained. It is shown that, as we refine the parametrization, the solution sequence of the approximate problems converges to the solution of the infinite programming problem (hence, to the solution of the original optimal control problem). Numerically, the semi-infinite programming problems obtained above can be solved efficiently using an algorithm based on a dual parametrization method.

Optimum window design by semi-infinite quadratic programming

This letter presents a new extended active set strategy for optimum finite impulse response (FIR) window design by semi-infinite quadratic programming. The windows may be asymmetric corresponding to frequency responses with general nonlinear phase. The optimality criterion is to minimize the sidelobe energy (L/sub 2/-norm) subject to a peak sidelobe magnitude-constraint (L/sub /spl infin//-constraint). Additional linear constraints are used to form the mainlobe (unity DC gain). Numerical examples involving group delay specifications are used to illustrate the usefulness of the algorithm.

Identification of Single-Input Multi-Output Nicely-Nonlinear Models

Abstract A system M is said to be a nicely nonlinear model of a given plant P if: i) M is tangent to P at the nominal equilibrium, ii) M is rigorously linearizable by output feedback. The present paper deals with the problem of identifying the parameters of a fairly general class of nicely nonlinear models. From a computational point of view, the problem to solve is a convex semi-infinite quadratic programming problem, for which a suitable algorithm has been set up.

On solving convex quadratic semi-infinite programming probelms

In this paper we study a from of convex quadratic semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. An entropic path-following algorithum is introduced with a convergence proof. Some practical implementations and numerical experiments are also included