SDP Duals without Duality Gaps for a Class of Convex Minimax Programs

In this paper we introduce a new dual program, which is representable as a semi-definite linear programming problem, for a primal convex minimax programming model problem and show that there is no duality gap between the primal and the dual whenever the functions involved are SOS-convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust SOS-convex programming problems under data uncertainty and to minimax fractional programming problems with SOS-convex polynomials. We obtain these results by first establishing sum of squares polynomial representations of non-negativity of a convex max function over a system of SOS-convex constraints. The new class of SOS-convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The SOS-convexity of polynomials can numerically be checked by solving semi-definite programming problems whereas numerically verifying convexity of polynomials is generally very hard

Robust Global Solutions of Bilevel Polynomial Optimization Problems with Uncertain Linear Constraints

This paper studies, for the first time, a bilevel polynomial program whose constraints involve uncertain linear constraints and another uncertain linear optimization problem. In the case of box data uncertainty, we present a sum of squares polynomial characterization of a global solution of its robust counterpart where the constraints are enforced for all realizations of the uncertainties within the prescribed uncertainty sets. By characterizing a solution of the robust counterpart of the lower-level uncertain linear program under spectrahedral uncertainty using a new generalization of Farkas' lemma, we reformulate the robust bilevel program as a single level non-convex polynomial optimization problem. We then characterize a global solution of the single level polynomial program by employing Putinar's Positivstellensatz of algebraic geometry under coercivity of the polynomial objective function. Consequently, we show that the robust global optimal value of the bilevel program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. Numerical examples are given to show how the robust optimal value of the bilevel program can be calculated by solving semidefinite programming problems using the Matlab toolbox YALMIP

Hybrid Approaches to Image Coding: A Review

Nowadays, the digital world is most focused on storage space and speed. With the growing demand for better bandwidth utilization, efficient image data compression techniques have emerged as an important factor for image data transmission and storage. To date, different approaches to image compression have been developed like the classical predictive coding, popular transform coding and vector quantization. Several second generation coding schemes or the segmentation based schemes are also gaining popularity. Practically efficient compression systems based on hybrid coding which combines the advantages of different traditional methods of image coding have also been developed over the years. In this paper, different hybrid approaches to image compression are discussed. Hybrid coding of images, in this context, deals with combining two or more traditional approaches to enhance the individual methods and achieve better-quality reconstructed images with higher compression ratio. Literature on hybrid techniques of image coding over the past years is also reviewed. An attempt is made to highlight the neuro-wavelet approach for enhancing coding efficiency.Comment: 7 pages, 3 figure

An Energy Efficient Neighbour Node Discovery Method for Wireless Sensor Networks

The discovery of neighbouring nodes in multihop wireless networks has become a key challenge. Due to tribulations in communication, synchronization loss between nodes, disparity in transmission power etc, the connectivity of nodes will always experience disruptions. On the other hand, the energy utilization by the nodes also became critical . In this paper, we propose a new method for neighbour discovery in wireless sensor networks (WSNs) which pays an eminent consideration for energy utilization and QoS parameters like latency, throughput, error rate etc. In the proposed method, the network routing is enhanced using AOMDV protocol which can accurately discover the neighbour nodes and power management with HMAC protocol which reduces the energy utilization significantly. A complete analysis is being performed to estimate how the Q o S metrics varies in various scenarios of power consumption in wireless networks.Comment: 4 figures and 5 page

Time Complexity Analysis of Binary Space Partitioning Scheme for Image Compression

Segmentation-based image coding methods provide high compression ratios when compared with traditional image coding approaches like the transform and sub band coding for low bit-rate compression applications. In this paper, a segmentation-based image coding method, namely the Binary Space Partition scheme, that divides the desired image using a recursive procedure for coding is presented. The BSP approach partitions the desired image recursively by using bisecting lines, selected from a collection of discrete optional lines, in a hierarchical manner. This partitioning procedure generates a binary tree, which is referred to as the BSP-tree representation of the desired image. The algorithm is extremely complex in computation and has high execution time. The time complexity of the BSP scheme is explored in this work.Comment: 5 pages, 5 figures, 2 tables, International Journal of Engineering and Innovative Technology; ISSN: 2277-3754 ISO 9001:200

Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that, for convex polynomial optimization, the Lasserre hierarchy with a slightly extended quadratic module always converges asymptotically even in the face of non-compact semi-algebraic feasible sets. We do this by exploiting a coercivity property of convex polynomials that are bounded below. We further establish that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point (rather than the objective function at each minimizer) guarantees finite convergence of the hierarchy. We obtain finite convergence by first establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets under a saddle-point condition. We finally prove that the existence of a saddle-point of the Lagrangian for a convex polynomial program is also necessary for the hierarchy to have finite convergence.Comment: 17 page

Robust SOS-Convex Polynomial Programs: Exact SDP Relaxations

This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes robust quadratically constrained convex programs and robust separable convex polynomial programs. It establishes sums of squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex programs under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncertainty sets, that allow second-order cone re-formulations of robust quadratically constrained programs, continue to permit exact SDP-relaxations for a broad class of robust SOS-convex programs. They also yield exact second-order cone relaxation for robust quadratically constrained programs

Convexifiability of Continuous and Discrete Nonnegative Quadratic Programs for Gap-Free Duality

In this paper we show that a convexifiability property of nonconvex quadratic programs with nonnegative variables and quadratic constraints guarantees zero duality gap between the quadratic programs and their semi-Lagrangian duals. More importantly, we establish that this convexifiability is hidden in classes of nonnegative homogeneous quadratic programs and discrete quadratic programs, such as mixed integer quadratic programs, revealing zero duality gaps. As an application, we prove that robust counterparts of uncertain mixed integer quadratic programs with objective data uncertainty enjoy zero duality gaps under suitable conditions. Various sufficient conditions for convexifiability are also given

Step and Search Control Method to Track the Maximum Power in Wind Energy Conversion Systems A Study

A simple step and search control strategy for extracting maximum output power from grid connected Variable Speed Wind Energy Conversion System (VSWECS) is implemented in this work. This system consists of a variable speed wind turbine coupled to a Permanent Magnet Synchronous Generator (PMSG) through a gear box, a DC-DC boost converter and a hysteresis current controlled Voltage Source Converter (VSC). The Maximum Power Point Tracking (MPPT) extracts maximum power from the wind turbine from cut-into rated wind velocity by sensing only by DC link power. This system can be connected to a micro-grid. Also it can be used for supplying an isolated local load by means of converting the output of Permanent Magnet Synchronous Generator (PMSG) to DC and then convert to AC by means of hysteresis current controlled Voltage Source Converter (VSI).Comment: 7 pages and 8 figure

Exact Conic Programming Reformulations of Two-Stage Adjustable Robust Linear Programs with New Quadratic Decision Rules

In this paper we introduce a new parameterized Quadratic Decision Rule (QDR), a generalisation of the commonly employed Affine Decision Rule (ADR), for two-stage linear adjustable robust optimization problems with ellipsoidal uncertainty and show that (affinely parameterized) linear adjustable robust optimization problems with QDRs are numerically tractable by presenting exact semi-definite program (SDP) and second order cone program (SOCP) reformulations. Under these QDRs, we also establish that exact conic program reformulations also hold for two-stage linear ARO problems, containing also adjustable variables in their objective functions. We then show via numerical experiments on lot-sizing problems with uncertain demand that adjustable robust linear optimization problems with QDRs improve upon the ADRs in their performance both in the worst-case sense and after simulated realization of the uncertain demand relative to the true solution.Comment: 19 pages, 4 figure