Complementarity and related problems

In this thesis, we present results related to complementarity problems. We study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We present algorithms for this problem, and exemplify it by a numerical example. Following this result, we explore the stochastic version of this linear complementarity problem. Finally, we apply complementarity problems on extended second order cones in a portfolio optimisation problem. In this application, we exploit our theoretical results to find an analytical solution to a new portfolio optimisation model. We also study the spherical quasi-convexity of quadratic functions on spherically self-dual convex sets. We start this study by exploring the characterisations and conditions for the spherical positive orthant. We present several conditions characterising the spherical quasi-convexity of quadratic functions. Then we generalise the conditions to the spherical quasi-convexity on spherically self-dual convex sets. In particular, we highlight the case of spherical second order cones

Advertising strategy for profit-maximization: a novel practice on Tmall's online ads manager platforms

Ads manager platform gains popularity among numerous e-commercial vendors/advertisers. It helps advertisers to facilitate the process of displaying their ads to target customers. One of the main challenges faced by advertisers, especially small and medium-sized enterprises, is to configure their advertising strategy properly. An ineffective advertising strategy will bring too many ``just looking'' clicks and, eventually, generate high advertising expenditure unproportionally to the growth of sales. In this paper, we present a novel profit-maximization model for online advertising optimization. The optimization problem is constructed to find optimal set of features to maximize the probability that target customers buy advertising products. We further reformulate the optimization problem to a knapsack problem with changeable parameters, and introduce a self-adjusted algorithm for finding the solution to the problem. Numerical experiment based on statistical data from Tmall show that our proposed method can optimize the advertising strategy given expenditure budget effectively.Comment: Online advertising campaign

Error bound and exact penalty method for optimization problems with nonnegative orthogonal constraint

This paper is concerned with a class of optimization problems with the nonnegative orthogonal constraint, in which the objective function is $L$-smooth on an open set containing the Stiefel manifold ${\rm St}(n,r)$. We derive a locally Lipschitzian error bound for the feasible points without zero rows when $n>r>1$, and when $n>r=1$ or $n=r$ achieve a global Lipschitzian error bound. Then, we show that the penalty problem induced by the elementwise $\ell_1$-norm distance to the nonnegative cone is a global exact penalty, and so is the one induced by its Moreau envelope under a lower second-order calmness of the objective function. A practical penalty algorithm is developed by solving approximately a series of smooth penalty problems with a retraction-based nonmonotone line-search proximal gradient method, and any cluster point of the generated sequence is shown to be a stationary point of the original problem. Numerical comparisons with the ALM \citep{Wen13} and the exact penalty method \citep{JiangM22} indicate that our penalty method has an advantage in terms of the quality of solutions despite taking a little more time.Comment: 34 pages, and 6 figure

A relaxation method for binary orthogonal optimization problems with its applications

This paper focuses on a class of binary orthogonal optimization problems frequently arising in semantic hashing. Consider that this class of problems may have an empty feasible set, rendering them not well-defined. We introduce an equivalent model involving a restricted Stiefel manifold and a matrix box set, and then investigate its penalty problems induced by the $\ell_1$-distance from the box set and its Moreau envelope. The two penalty problems are always well-defined, and moreover, they serve as the global exact penalties provided that the original model is well-defined. Notably, the penalty problem induced by the Moreau envelope is a smooth optimization over an embedded submanifold with a favorable structure. We develop a retraction-based nonmonotone line-search Riemannian gradient method to address this penalty problem to achieve a desirable solution for the original binary orthogonal problems. Finally, the proposed method is applied to supervised and unsupervised hashing tasks and is compared with several popular methods on the MNIST and CIFAR-10 datasets. The numerical comparisons reveal that our algorithm is significantly superior to other solvers in terms of feasibility violation, and it is comparable even superior to others in terms of evaluation metrics related to the Hamming distance.Comment: Binary orthogonal optimization problems, global exact penalty, relaxation methods, semantic hashin

An inexact linearized proximal algorithm for a class of DC composite optimization problems and applications

This paper is concerned with a class of DC composite optimization problems which, as an extension of the convex composite optimization problem and the DC program with nonsmooth components, often arises from robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) which in each step computes an inexact minimizer of a strongly convex majorization constructed by the partial linearization of their objective functions. The generated iterate sequence is shown to be convergent under the Kurdyka-{\L}ojasiewicz (KL) property of a potential function, and the convergence admits a local R-linear rate if the potential function has the KL property of exponent $1/2$ at the limit point. For the latter assumption, we provide a verifiable condition by leveraging the composite structure, and clarify its relation with the regularity used for the convex composite optimization. Finally, the proposed iLPA is applied to a robust factorization model for matrix completions with outliers, DC programs with nonsmooth components, and $\ell_1$-norm exact penalty of DC constrained programs, and numerical comparison with the existing algorithms confirms the superiority of our iLPA in computing time and quality of solutions