Global Maximizer Research Articles

An Extreme-Point Subdifferential Method for Convex Hull Pricing in Energy and Reserve Markets—Part I: Algorithm Structure

Prices in electricity markets are given by the dual variables associated with the supply-demand constraint in the dispatch problem. However, in unit-commitment-based day-ahead markets, these variables are not easy to obtain. A common approach relies on re-solving the dispatch problem with the commitment decisions fixed, and utilizing the associated dual variables. This avenue may lead to inadequate revenues to generators, which has led to the introduction of uplift payments made by the market operator for further compensating the generators. An alternative pricing mechanism known as convex hull pricing has been proposed to reduce or eliminate uplift payments. Computation of these prices requires the global maximization of an associated Lagrangian dual problem. In this paper, we present an extreme-point-based procedure for obtaining a global maximizer. Unlike standard subgradient schemes where an arbitrary subgradient is used, we present an extreme-point subdifferential (EPSD) algorithm; this is a novel technique in which the steepest ascent direction is constructed by solving a continuous quadratic program. The EPSD algorithm initiates a move along this direction, employing an a priori constant steplength, with the intent of reaching the boundary of the face. A backtracking scheme selects a steplength that ensures descent with respect to a suitably defined merit function. As most electricity markets today co-optimize energy and reserves, an extension of the proposed convex hull pricing algorithm is provided for such integrated markets. Under suitable assumptions, we compare outcomes of energy-only and energy-reserve co-optimized markets under different pricing and uplift rules. In these examples, pricing rules have a major impact on the total payment while the uplift payment only accounts for a small portion of it. We also observe that it remains unclear whether marginal-cost pricing or convex-hull pricing leads to higher total payment.

A note on the trace quotient problem

The trace quotient problem or the trace ratio problem (TRP) is to find an orthogonal matrix \(V\in \mathbb{R }^{m\times \ell } ~(m\ge \ell )\) that maximizes the quotient \({{\mathrm{tr}}}(V^\top BV)/{{\mathrm{tr}}}(V^\top WV)\) for a given symmetric matrix \(B\in \mathbb R ^{m\times m}\) and a symmetric positive definite matrix \(W\in \mathbb R ^{m\times m}.\) It has a crucial role in linear discriminant analysis and has many other applications in computer vision and machine learning as well. In this short note, we first establish the classical first and second order optimality conditions for TRP. As a straightforward application of these optimality conditions, we contribute a simple proof for the property that TRP does not admit local non-global maximizer, which is first proved by Shen et al. (A geometric revisit to the trace quotient problem, proceedings of the 19th International Symposium of Mathematical Theory of Networks and Systems, 2010) based on Grassmann manifold. Without involving much knowledge of the underlying differential geometry, our proof primarily uses basic properties in linear algebra, which also leads to an effective starting pointing strategy for any monotonically convergent iteration to find the global maximizer of TRP.

Parameter estimation from load-sharing system data using the expectation–maximization algorithm

This article considers a system of multiple components connected in parallel. As components fail one by one, the remaining working components share the total load applied to the system. This is commonly referred to as load sharing in the reliability engineering literature. This article considers the traditional approach to the modeling of a load-sharing system under the assumption of the existence of underlying hypothetical latent random variables. Using the Expectation–Maximization (EM) algorithm, a methodology is proposed to obtain the maximum likelihood estimates in such a model in the case where the underlying lifetime distribution of the components is lognormal or normal. The proposed EM method is also illustrated and substantiated using numerical examples. The estimates obtained using the EM algorithm are compared with those obtained using the Broyden–Fletcher–Goldfarb–Shanno algorithm, which falls under the class of numerical methods known as Newton or quasi-Newton methods. The results show that the estimates obtained using the proposed EM method always converge to a unique global maximizer, whereas the estimates obtained using the Newton-type method are highly sensitive to the choice of starting values and thus often fail to converge.

Triangular norm based graded convex fuzzy sets

In this paper, we develop the theory of graded convex fuzzy set, graded strongly convex fuzzy set, graded strictly convex fuzzy set based on triangular norms, in which the degree of fuzzy convexity is described by one real number. We also investigate some graded properties of these graded convex fuzzy sets and present the relations between these graded convex fuzzy sets and their topological properties and give several conditions for a fuzzy set to be a graded convex fuzzy set. Finally, we generalize the notion of global maximizer and prove an important property of graded convex fuzzy sets.

A discrete dynamic convexized method for the max-cut problem

The max-cut problem is a classical NP-hard problem in graph theory. In this paper, we adopt a local search method, called MCFM, which is a simple modification of the Fiduccia-Mattheyses heuristic method in Fiduccia and Mattheyses (Proc. ACM/IEEE DAC, pp. 175–181, 1982) for the circuit partitioning problem in very large scale integration of circuits and systems. The method uses much less computational cost than general local search methods. Then, an auxiliary function is presented which has the same global maximizers as the max-cut problem. We show that maximization of the function using MCFM can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. An algorithm is proposed for the max-cut problem, by maximizing the auxiliary function using MCFM from random initial solutions. Computational experiments were conducted on three sets of standard test instances from the literature. Experimental results show that the proposed algorithm is effective for the three sets of standard test instances.

On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere

Given symmetric matrices B,D?? n×n and a symmetric positive definite matrix W?? n×n , maximizing the sum of the Rayleigh quotient x ? D x and the generalized Rayleigh quotient $\frac{\mathbf{x}^{\top}B \mathbf{x}}{\vphantom{\mathrm{I}^{\mathrm{I}}}\mathbf{x}^{\top}W\mathbf{x}}$ on the unit sphere not only is of mathematical interest in its own right, but also finds applications in practice. In this paper, we first present a real world application arising from the sparse Fisher discriminant analysis. To tackle this problem, our first effort is to characterize the local and global maxima by investigating the optimality conditions. Our results reveal that finding the global solution is closely related with a special extreme nonlinear eigenvalue problem, and in the special case D=μW (μ>0), the set of the global solutions is essentially an eigenspace corresponding to the largest eigenvalue of a specially-defined matrix. The characterization of the global solution not only sheds some lights on the maximization problem, but motives a starting point strategy to obtain the global maximizer for any monotonically convergent iteration. Our second part then realizes the Riemannian trust-region method of Absil, Baker and Gallivan (Found. Comput. Math. 7:303---330, 2007) into a practical algorithm to solve this problem, which enjoys the nice convergence properties: global convergence and local superlinear convergence. Preliminary numerical tests are carried out and empirical evaluation of its performance is reported.

Ergodic Secrecy Rate for Multiple-Antenna Wiretap Channels With Rician Fading

A Gaussian multiple-antenna wiretap channel model is considered, where there exists a transmitter equipped with multiple antennas, a legitimate receiver equipped with a single antenna, and an eavesdropper equipped with multiple antennas. We study the problem of finding the optimal input covariance that maximizes the ergodic secrecy rate subject to a power constraint, assuming that full information on the legitimate channel is available to the transmitter, but only statistical information on the eavesdropper channel is known. More specifically, we investigate the case of MIMO Rician fading for the eavesdropper channel. We show that the optimal input covariance has rank one, which allows us to reduce the original optimization problem to a smooth one variable optimization problem. We propose a Newton-type method for a local maximizer, and Piyavskii's algorithm for the global maximizer. Numerical results are presented to illustrate the proposed algorithms.

An alternating variable method for the maximal correlation problem

The maximal correlation problem (MCP) aiming at optimizing correlations between sets of variables plays an important role in many areas of statistical applications. Up to date, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem (MEP), which serves only as a necessary condition for the global maxima of the MCP. For statistical applications, the global maximizer is quite desirable. In searching the global solution of the MCP, in this paper, we propose an alternating variable method (AVM), which contains a core engine in seeking a global maximizer. We prove that (i) the algorithm converges globally and monotonically to a solution of the MEP, (ii) any convergent point satisfies a global optimal condition of the MCP, and (iii) whenever the involved matrix A is nonnegative irreducible, it converges globally to the global maximizer. These properties imply that the AVM is an effective approach to obtain a global maximizer of the MCP. Numerical testings are carried out and suggest a superior performance to the others, especially in finding a global solution of the MCP.

Global optimization design method for maximizing the capacity of V-belt drive

An optimization model is constructed to formulate the maximization problem on the capacity of V-belt drive. The concavity, the monotonicity and the global optimality condition are studied for the objective function, and it is proved that the feasible region of the model is bounded, closed and convex under some design conditions. Then, a solution method, called an optimal segment algorithm, is developed to find the global maximizer of the model. Under four different design conditions, solution methods are presented respectively. Some real case studies are employed to demonstrate that the model and the algorithm in this paper are promising.

Iterative Shrinkage Approach to Restoration of Optical Imagery

The problem of reconstruction of digital images from their degraded measurements is regarded as a problem of central importance in various fields of engineering and imaging sciences. In such cases, the degradation is typically caused by the resolution limitations of an imaging device in use and/or by the destructive influence of measurement noise. Specifically, when the noise obeys a Poisson probability law, standard approaches to the problem of image reconstruction are based upon using fixed-point algorithms which follow the methodology first proposed by Richardson and Lucy. The practice of using these methods, however, shows that their convergence properties tend to deteriorate at relatively high noise levels. Accordingly, in the present paper, a novel method for denoising and/or deblurring of digital images corrupted by Poisson noise is introduced. The proposed method is derived under the assumption that the image of interest can be sparsely represented in the domain of a linear transform. Consequently, a shrinkage-based iterative procedure is proposed, which guarantees the solution to converge to the global maximizer of an associated maximum a posteriori criterion. It is shown in a series of computer-simulated experiments that the proposed method outperforms a number of existing alternatives in terms of stability, precision, and computational efficiency.

Towards the global solution of the maximal correlation problem

The maximal correlation problem (MCP) aiming at optimizing correlation between sets of variables plays a very important role in many areas of statistical applications. Currently, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem for a related matrix A, which serves as a necessary condition for the global solutions of the MCP. However, the reliability of the statistical prediction in applications relies greatly on the global maximizer of the MCP, and would be significantly impacted if the solution found is a local maximizer. Towards the global solution of the MCP, we have obtained four results in the present paper. First, the sufficient and necessary condition for global optimality of the MCP when A is a positive matrix is extended to the nonnegative case. Secondly, the uniqueness of the multivariate eigenvalues in the global maxima of the MCP is proved either when there are only two sets of variables involved, or when A is nonnegative. The uniqueness of the global maximizer of the MCP for the nonnegative irreducible case is also proved. These theoretical achievements lead to our third result that if A is a nonnegative irreducible matrix, both the Horst-Jacobi algorithm and the Gauss-Seidel algorithm converge globally to the global maximizer of the MCP. Lastly, some new estimates of the multivariate eigenvalues related to the global maxima are obtained.

Finding the Maximizers of the Information Divergence From an Exponential Family

This paper investigates maximizers of the information divergence from an exponential family $E$. It is shown that the $rI$-projection of a maximizer $P$ to $E$ is a convex combination of $P$ and a probability measure $P_-$ with disjoint support and the same value of the sufficient statistics $A$. This observation can be used to transform the original problem of maximizing $D(\cdot||E)$ over the set of all probability measures into the maximization of a function $\Dbar$ over a convex subset of $\ker A$. The global maximizers of both problems correspond to each other. Furthermore, finding all local maximizers of $\Dbar$ yields all local maximizers of $D(\cdot||E)$. This paper also proposes two algorithms to find the maximizers of $\Dbar$ and applies them to two examples, where the maximizers of $D(\cdot||E)$ were not known before.

On image reconstruction algorithms for binary electromagnetic geotomography

In this paper we discuss the selected image reconstruction methods of binary tomography in the context of their application to geophysical imaging. We restrict our considerations to a discrete version of high-frequency electromagnetic geotomography, which we label as Binary Electromagnetic Geotomography (BEG). Basically, such an imaging technique may be applied to detect subsurface anomalies (air-filled voids) whose attenuation coefficient is very low (nearly zero-value) and considerably different from that for the background. The assumption for a binary representation of the image to be reconstructed substantially relaxes image reconstruction problems related to ill-posedness that comes from an intrinsic limitation of an angular range of projections. We test two algorithms for binary tomography, where the penalty term is based on the Markov Random Field (MRF) model. The mean-field reference distribution and mean-field annealing are applied to estimate the global maximizer of the Gibbs–Boltzmann distribution associated with the objective function. We also apply the projected gradient algorithm that uses a binary steering. Very efficient implementations of the algorithms are also given. The numerical results are presented for noise-free, noisy, and real data.

Energy extremals and nonlinear stability in a variational theory of a coupled barotropic flow - Rotating solid sphere system

A new variational principle—extremizing the fixed frame kinetic energy under constant relative enstrophy—for a coupled barotropic flow—rotating solid sphere system is introduced with the following consequences. In particular, angular momentum is transferred between the fluid and the solid sphere through a modeled torque mechanism. The fluid’s angular momentum is therefore not fixed but only bounded by the relative enstrophy, as is required of any model that supports super-rotation. The main results are: At any rate of spin Ω and relative enstrophy, the unique global energy maximizer for fixed relative enstrophy corresponds to solid-body super-rotation; the counter-rotating solid-body flow state is a constrained energy minimum provided the relative enstrophy is small enough, otherwise, it is a saddle point. For all energy below a threshold value which depends on the relative enstrophy and solid spin Ω, the constrained energy extremals consist of only minimizers and saddles in the form of counter-rotating states. Only when the energy exceeds this threshold value can pro-rotating states arise as global maximizers. Unlike the standard barotropic vorticity model which conserves angular momentum of the fluid, the counter-rotating state is rigorously shown to be nonlinearly stable only when it is a local constrained minima. The global constrained maximizer corresponding to super-rotation is always nonlinearly stable.

Further investigation of abstract convexity with respect to the class of general min-type functions

In this article we continue the examination of the basic concepts of abstract convexity, taking for the set of elementary functions the totality of the so-called min-type functions which are defined on the n-dimensional Euclidean space Rn as the pointwise minimum of finite collections of linear functions. We examine compact -convex sets, -convex hull and -extreme points of a subset of . We show that among global maximizers of a convex-along-rays function over an -convex set there are -extreme points of the set. We introduce the positive and the negative polar sets and examine their properties. We present the description of the bipolar sets and show that a subset of is -convex if it is the intersection of its positive and negative bipolar sets. We describe the support set of the upper envelope of a subset of .

Testing for a Global Maximum of the Likelihood

When several roots to the likelihood equation exist, the root corresponding to the global maximizer of the likelihood is generally retained. This procedure, however, supposes that all possible roots are identified. Because in many cases the global maximizer is the only consistent root, we propose a test to detect if a given solution is consistent. This test relies on some necessary and sufficient conditions for consistency of a root and simply consists of comparing the difference between two expected log-likelihood expressions. Monte Carlo studies and a real life example show that the proposed procedure leads to encouraging results. In particular, it clearly outperforms another available test of this kind, especially for relatively small sample sizes.

Preincavity and fuzzy decision making

Since almost all practical problems are fuzzy and approximate, fuzzy decision making becomes one of the most important practical approaches. However, the resulting problems are frequently complicated and difficult to solve. One effective way to overcome these difficulties is to explore the concavity or generalized concavity properties of the resulting problems. In this paper, we introduce and study the concept of supp-preincave and supp-prequasiincave fuzzy sets. We give characterizations for a supp-preincave fuzzy set in terms of its fuzzy hypograph, and a supp-prequasiincave fuzzy set in terms of its level sets. Furthermore, we also prove that any local maximizer of a supp-preincave fuzzy set is also a global maximizer, and that any strictly local maximizer of a supp-prequasiincave fuzzy set is also a global maximizer. Finally, some aggregation rules for supp-preincave and supp-prequasiincave fuzzy sets are given and some applications to fuzzy decision making are discussed.

On complexity of unconstrained hyperbolic 0–1 programming problems

Single- and multiple-ratio unconstrained hyperbolic 0–1 programming problems are considered. We prove that checking whether these problems have a unique solution is NP-hard. Furthermore, we show that finding the global maximizer of problems with unique solution remains NP-hard. We also discuss complexity of local search and approximability for multiple-ratio problems.

Fuzzy Φ-convexity and fuzzy decision making

The notion of convexity and its various generalization is important for quantitative and qualitative studies in operations research or applied mathematics. It has also been considered by many authors in fuzzy set theory. In this paper, we study the concept of Φ-convex and Φ-quasiconvex fuzzy sets which was proposed byChen et al. [1], and develop some useful extrema properties of these fuzzy sets. We prove that any local maximizer of a Φ-convex fuzzy set is also a global maximizer, and that any strict local maximizer of a Φ-quasiconvex fuzzy set is also a global maximizer. We also study the class of strictly Φ-convex (respectively, strictly Φ-quasiconvex) fuzzy sets that is more restricted than the class of Φ-convex (respectively, Φ-quasiconvex) fuzzy sets. We prove for both families of strictly Φ-convex and strictly Φ-quasiconvex fuzzy sets that every local maximizer is also the unique global maximizer. In addition, some applications to fuzzy decision making are discussed.

EM-Based Multiuser Detection in Fast Fading Multipath Environments

We address the problem of multiuser detection in fast fading multipath environments for DS-CDMA systems. In fast fading scenarios, temporal variations of the channel cause significant performance degradation even with the Rake receiver. We use a previously introduced time-frequency (TF) Rake receiver based on a canonical formulation of the channel and signals to simultaneously combat fading and multipath effects. This receiver uses the Doppler spread caused by rapid time-varying channel as another means of diversity. In dealing with multiaccess interference and as an attempt to avoid the prohibitive computational complexity of the optimum maximum-likelihood (ML) detector, we use the expectation maximization (EM) algorithm to derive an approximate ML detector. The new detector turns out to have an iterative structure very similar to the well-known multistage detector with some extra parameters. At the two extreme values of these parameters, the EM detector reduces to either one-shot TF Rake or generalized multistage detector. For the intermediate values of the parameters, it combines the two estimates to obtain a better decision for the bits of the users. Because of using the EM algorithm, this detector has better convergence properties than the multistage detector; the bit estimates always converge, and if an appropriate initial vector is used, they converge to the global maximizer of the likelihood function. As a result, the new detector provides significantly improved performance while maintaining the low complexity of the multistage detector. Our simulation results confirm the expected performance improvements compared to the base case of the TF Rake as well as the multistage detector used with the TF Rake.