Optimality Conditions and Duality for Semi-Infinite Mathematical Programming Problem with Equilibrium Constraints

This article considers a semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC) defined as a semi-infinite mathematical programming problem with complementarity constraints. We establish necessary and sufficient optimality conditions for the (SIMPEC). We also formulate Wolfe- and Mond-Weir-type dual models for (SIMPEC) and establish weak, strong and strict converse duality theorems for (SIMPEC) and the corresponding dual problems under invexity assumptions.

GLOBAL REGISTRATION OF MULTIPLE POINT CLOUDS USING SEMIDEFINITE PROGRAMMING

Consider N points in R-d and M local coordinate systems that are related through unknown rigid transforms. For each point, we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The least-squares formulation of this problem, although nonconvex, has a well-known closed-form solution when M = 2 (based on the singular value decomposition (SVD)). However, no closed-form solution is known for M >= 3. In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely, a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results from rigidity theory, we prove conditions for exact and stable recovery for the SDP relaxation. In particular, we prove that the SDP relaxation can guarantee recovery under more adversarial conditions compared to earlier proposed spectral relaxations, and we derive error bounds for the registration error incurred by the SDP relaxation. We also present results of numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the SDP (i.e., we are able to solve the original nonconvex least-squares problem) up to a certain noise threshold, and (b) the SDP performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.

Optimal Power Allocation Policies for Low Powered Wireless Communication Systems

We consider optimal average power allocation policies in a wireless channel in the presence of individual delay constraints on the transmitted packets. Power is consumed in transmission of data only. We consider the case when the power used in transmission is a linear function of the data transmitted. The transmission channel may experience multipath fading. We have developed a computationally efficient online algorithm, when there is same hard delay constraint for all packets. Later on, we generalize it to the case when there are multiple real time streams with different hard deadline constraints. Our algorithm uses linear programming and has very low complexity.

MIMO Decode-and-Forward Relay Beamforming for Secrecy with Cooperative Jamming

In this paper, we consider decode-and-forward (DF) relay beamforming for secrecy with cooperative jamming (CJ) in the presence of multiple eavesdroppers. The communication between a source-destination pair is aided by a multiple-input multiple-output (MIMO) relay. The source has one transmit antenna and the destination and eavesdroppers have one receive antenna each. The source and the MIMO relay are constrained with powers P-S and P-R, respectively. We relax the rank-1 constraint on the signal beamforming matrix and transform the secrecy rate max-min optimization problem to a single maximization problem, which is solved by semidefinite programming techniques. We obtain the optimum source power, signal relay weights, and jamming covariance matrix. We show that the solution of the rank-relaxed optimization problem has rank-1. Numerical results show that CJ can improve the secrecy rate.

MIMO Decode-and-Forward Relay Beamforming for Secrecy with Cooperative Jamming

In this paper, we consider decode-and-forward (DF) relay beamforming for secrecy with cooperative jamming (CJ) in the presence of multiple eavesdroppers. The communication between a source-destination pair is aided by a multiple-input multiple-output (MIMO) relay. The source has one transmit antenna and the destination and eavesdroppers have one receive antenna each. The source and the MIMO relay are constrained with powers P-S and P-R, respectively. We relax the rank-1 constraint on the signal beamforming matrix and transform the secrecy rate max-min optimization problem to a single maximization problem, which is solved by semidefinite programming techniques. We obtain the optimum source power, signal relay weights, and jamming covariance matrix. We show that the solution of the rank-relaxed optimization problem has rank-1. Numerical results show that CJ can improve the secrecy rate.

Folivory as a Constraint on Social Behaviour of Langurs in South India

Folivory, being a dietary constraint, can affect the social time of colobines. In the present study, we compared food items and activity budgets of two closely related species of colobines inhabiting South India, i.e. the Hanuman langur (Semnopithecus hypoleucos) and Nilgiri langur (Semnopithecus johnii), to determine whether folivory had an impact on social time in these species. Our study established that Nilgiri langurs were more folivorous than Hanuman langurs. Nilgiri langurs spent much less time on social activities, but more time on resting, although the social organization of S. hypoleucos was similar to that of the Nilgiri langur. The enforced resting time for fermentation of leafy food items may have reduced the time available for social interactions, which in turn affected the social time in Nilgiri langurs. By comparing the data from previous studies on other Hanuman langur species, we found that S. hypoleucos spent a similar amount of time on social activities as Semnopithecus entellus. Hence, the social behaviour of S. entellus and S. hypoleucos is phylogenetically highly conservative. (C) 2015 S. Karger AG, Basel

Sum Secrecy Rate in Multicarrier DF Relay Beamforming

In this paper, we study sum secrecy rate in multicarrier decode-and-forward relay beamforming. We obtain the optimal source power and relay weights on each subcarrier which maximize the sum secrecy rate. For a given total power on a given subcarrier k, P-0(k), we reformulate the optimization problem by relaxing the rank-1 constraint on the complex positive semidefinite relay weight matrix, and solve using semidefinite programming. We analytically prove that the solution to the relaxed optimization problem is indeed rank 1. We show that the subcarrier secrecy rate, R-s (P-0(k)), is a concave function in total power P-0(k) if R-s (P-0(k)) > 0 for any P-0(k) > 0. Numerical results show that the sum secrecy rate with optimal power allocation across subcarriers is more than the sum secrecy rate with equal power allocation. We also propose a low complexity suboptimal power allocation scheme which outperforms equal power allocation scheme.

Optimal Guidance for Accurate Lunar Soft Landing with Minimum Fuel Consumption using Model Predictive Static Programming

In this paper the soft lunar landing with minimum fuel expenditure is formulated as a nonlinear optimal guidance problem. The realization of pinpoint soft landing with terminal velocity and position constraints is achieved using Model Predictive Static Programming (MPSP). The high accuracy of the terminal conditions is ensured as the formulation of the MPSP inherently poses final conditions as a set of hard constraints. The computational efficiency and fast convergence make the MPSP preferable for fixed final time onboard optimal guidance algorithm. It has also been observed that the minimum fuel requirement strongly depends on the choice of the final time (a critical point that is not given due importance in many literature). Hence, to optimally select the final time, a neural network is used to learn the mapping between various initial conditions in the domain of interest and the corresponding optimal flight time. To generate the training data set, the optimal final time is computed offline using a gradient based optimization technique. The effectiveness of the proposed method is demonstrated with rigorous simulation results.

Image deconvolution using a Gaussian scale mixtures model to approximate the wavelet sparseness constraint

This paper proposes to use an extended Gaussian Scale Mixtures (GSM) model instead of the conventional ℓ1 norm to approximate the sparseness constraint in the wavelet domain. We combine this new constraint with subband-dependent minimization to formulate an iterative algorithm on two shift-invariant wavelet transforms, the Shannon wavelet transform and dual-tree complex wavelet transform (DTCWT). This extented GSM model introduces spatially varying information into the deconvolution process and thus enables the algorithm to achieve better results with fewer iterations in our experiments. ©2009 IEEE.

Optimal bounded control of first-passage failure of strongly non-linear oscillators under combined harmonic and white-noise excitations

A procedure for designing the optimal bounded control of strongly non-linear oscillators under combined harmonic and white-noise excitations for minimizing their first-passage failure is proposed. First, a stochastic averaging method for strongly non-linear oscillators under combined harmonic and white-noise excitations using generalized harmonic functions is introduced. Then, the dynamical programming equations and their boundary and final time conditions for the control problems of maximizing reliability and of maximizing mean first-passage time are formulated from the averaged Ito equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraint. Finally, the conditional reliability function, the conditional probability density and mean of the first-passage time of the optimally controlled system are obtained from solving the backward Kolmogorov equation and Pontryagin equation. An example is given to illustrate the proposed procedure and the results obtained are verified by using those from digital simulation. (C) 2003 Elsevier Ltd. All rights reserved.