We consider maximum likelihood estimation (MLE) of heteroscedastic regression models based on a new “parametrization” of the likelihood in terms of the Sharpe ratio function, or the ratio of the mean and volatility functions. While with a standard parametrization the MLE problem is not convex and hence hard to solve globally, our parametrization leads to a functional that is jointly convex in the Sharpe ratio and inverse volatility functions. The major difficulty with the resulting infinite-dimensional convex program is the shape constraint on the inverse volatility function. We propose to solve the problem by solving a sequence of finite-dimensional convex programs with increasing dimensions, which can be done globally and efficiently. We demonstrate that, when the goal is to estimate the Sharpe ratio function directly, the finite-sample performance of the proposed estimation method is superior to existing methods that estimate the mean and variance functions separately. When applied to a financial dataset, our method captures a well-known covariate-dependent effect on the Shape ratio.

Abstract-This article describes a minimax regret approach to robust beamforming with an ellipsoidal uncertainty model for a steering vector, in which the objective is to minimize the worstcase regret over the uncertainty set, where 'worst' means largest. This problem can be solved efficiently by using an iterative method which uses an alternating sequence of optimization and worst-case analysis steps. Each of the two steps amounts to solving a convex optimization problem. The method typically converges to a solution within 5 iterations. The minimax regret beamforming is illustrated with numerical examples in planar random array antennas. The numerical results show that the minimax regret approach is less pessimistic (less conservative) and provides more robust performance than the worst-case SINR maximization (maximin) approach, where 'worst' means smallest.

In detecting a deterministic signal in the presence of Gaussian noise, the receiver operating characteristic (ROC) curve determined by a linear detector with slope maximizing the signal-to-noise ratio (SNR) and with varying threshold characterizes limits of detection performance. In this note, we consider the problem of detecting a signal in the presence of uncertainty in the signal itself and the noise covariance. We show that the classical result can be generalized to robust signal detection with a convex uncertainty model: the ROC curve determined by a linear detector with slope maximizing the worst-case SNR gives limits of detection performance in the worst-case sense. The worst-case SNR maximization problem can be solved using convex optimization, so robust ROC analysis is tractable.

The performance of a kernel-based learning algorithm depends very much on the choice of the kernel. Recently, much attention has been paid to the problem of learning the kernel itself from given training examples. The main emphasis has been on formulating the problem as a tractable convex optimization problem. Only for a few very special cases such as support vector machines are explicit convex formulations known. In this paper, we show that, in a wide variety of kernel-based learning algorithms, the kernel learning problem can be formulated as a convex optimization problem which interior-point methods can solve globally and efficiently. The kernel learning method is illustrated with a regression problem that arises in petroleum engineering.

A fast and simple algorithm for designing time-optimal waveforms is presented. The algorithm accepts a given arbitrary multidimensional k-space trajectory as the input and outputs the time-optimal gradient waveform that traverses k-space along that path in minimum time. The algorithm is noniterative, and its run time is independent of the complexity of the curve, i.e., the number of switches between slew-rate limited acceleration, slew-rate limited deceleration, and gradient amplitude limited regions. The key in the method is that the gradient amplitude is designed as a function of arc length along the k-space trajectory, rather than as a function of time. Several trajectory design examples are presented.

In Chebyshev finite-impulse response (FIR) equalization, we design an FIR filter that minimizes the Chebyshev equalization error,i.e., the maximum absolute deviation between the equalized and the desired frequency response functions, assuming the unequalized response function is known exactly. In robust Chebyshev FIR equalization, we take into account uncertainty in the unequalized response function, described as a set of possible values for the unequalized response at each frequency, by designing an FIR filter that minimizes worst-case Chebyshev equalization error over all possible unequalized response functions. When the uncertainty in unequalized response function is described by a complex uncertainty ellipsoid, at each frequency, we show that the robust Chebyshev FIR equalization design problem can be formulated as a semidefinite program (SDP), and therefore efficiently (and globally) solved. When the uncertainty is given by a complex disk, the design problem can be formulated as a second-order cone program (SOCP), which can be solved almost as fast as the nominal Chebyshev equalization problem (ignoring uncertainty). The robust equalizer design method is demonstrated with a numerical example.

Compressed sensing or compressive sampling (CS) has been receiving a lot of interest as a promising method for signal recovery and sampling. CS problems can be cast as convex problems, and then solved by several standard methods such as interior-point methods, at least for small and medium size problems. In this paper we describe a specialized interior-point method for solving CS problems that uses a preconditioned conjugate gradient method to compute the search step. The method can efficiently solve large CS problems, by exploiting fast algorithms for the signal transforms used. The method is demonstrated with a medical resonance imaging (MRI) example.

In this letter, we show that worst-case robust beamforming, with uncertain weights subject to multiplicative variations, can be cast as a convex optimization problem. We interpret this problem as a weighted complex l1-regularization of the nominal beamforming problem, and show that it can be solved with the same computational complexity as nominal beamforming, ignoring the variations. We derive a simple lower bound on how much worse the robust beamformer will be compared to the nominal beamformer solution with no weight uncertainty. We demonstrate the robust approach with a simple narrowband beamformer

In Kernel Fisher discriminant analysis (KFDA), we carry out Fisher linear discriminant analysis in a high dimensional feature space defined implicitly by a kernel. The performance of KFDA depends on the choice of the kernel; in this paper, we consider the problem of finding the optimal kernel, over a given convex set of kernels. We show that this optimal kernel selection problem can be reformulated as a tractable convex optimization problem which interior-point methods can solve globally and efficiently. The kernel selection method is demonstrated with some UCI machine learning benchmark examples.

We consider the problem of choosing a linear classifier that minimizes misclassification probabilities in two-class classification, which is a bi-criterion problem, involving a trade-off between two objectives. We assume that the class-conditional distributions are Gaussian. This assumption makes it computationally tractable to find Pareto optimal linear classifiers whose classification capabilities are inferior to no other linear ones. The main purpose of this paper is to establish several robustness properties of those classifiers with respect to variations and uncertainties in the distributions. We also extend the results to kernel-based classification. Finally, we show how to carry out trade-off analysis empirically with a finite number of given labeled data.

This paper proposes a transmit power optimization algorithm for a Gaussian vector broadcast channel. The optimal power allocation minimizes the sum trace of input covariance matrices of all the users in the Gaussian vector broadcast channel. By duality the transmit power optimization can be transformed into an equivalent convex optimization in a multiple access channel. Recursive equations are derived as sufficient conditions for the minimum transmit power and an iterative algorithm is proposed to solve the equations. Numerical examples show that the iterative algorithm converges very fast and efficiently determines the input covariance matrices for the optimum power allocation.

This tutorial concerns a method for solving a variety of circuit sizing and optimization problems, which is based on formulating the problem as a geometric program (GP), or a generalized geometric program (GGP). These nonlinear, constrained optimization problems can be transformed to convex optimization problems, and then solved (globally) very efficiently.

Fisher linear discriminant analysis (LDA) can be sensitive to the problem data. Robust Fisher LDA can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a classification problem and optimizing for the worst-case scenario under this model. The main contribution of this paper is show that with general convex uncertainty models on the problem data, robust Fisher LDA can be carried out using convex optimization. For a certain type of product form uncertainty model, robust Fisher LDA can be carried out at a cost comparable to standard Fisher LDA. The method is demonstrated with some numerical examples. Finally, we show how to extend these results to robust kernel Fisher discriminant analysis, i.e., robust Fisher LDA in a high dimensional feature space.