A Practical Approach to Quasi-convex Optimization

Quasi-Relative Interiors for Graphs of Convex Set-Valued Mappings

This paper aims at providing further studies of the notion of quasi-relative interior for convex sets introduced by Borwein and Lewis. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.

Star subgradient projection for solving quasi-convex feasibility problems

In this work we consider an iterative method for solving the quasi-convex feasibility problem. We firstly introduce the so-called star subgradient projection operator and present some useful properties. We subsequently obtain a convergence result of the considered method by using properties of the introduced nonlinear operator.

Convergence Rates of Subgradient Methods for Quasi-convex Optimization Problems

Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we investigate the iteration complexity and convergence rates of various subgradient methods for solving quasi-convex optimization in a unified framework. In particular, we consider a sequence satisfying a general (inexact) basic inequality, and investigate the global convergence theorem and the iteration complexity when using the constant, diminishing or dynamic stepsi…

$X$-convexity and Applications of Quasi-$X$-Convex Functions

A class of real functions, which is the generalization of a family of convex functions, is introduced; in this connection, we have defined $X$-convex, strictly $X$-convex, quasi-$X$-convex, strictly quasi-$X$-convex, and semi-strictly quasi-$X$-convex functions. Moreover, in this paper, we give a detailed study of the fundamental properties of these functions with various examples, supporting the concepts. Finally, the study of optimization problems employs quasi-$X$-convex, semistrictly quasi-$X$-convex, and strictly quasi-$X$-convex functions.

关于测地凸集的支撑拟超平面与分离定理及其在 Riemannian 流形上的应用

On the supporting quasi-hyperplane and separation theorem of geodesic convex sets with applications on Riemannian manifolds

针对 Riemannian 流形上的局部测地凸集，研究证明点—集分离定理与边界支撑拟超平面的存在性，并据此导出带约束优化问题的最优性条件。

拟差凸性：拟可微优化的现代化

Quasi-difference-convexity: Modernization of Quasi-differentiable Optimization

本文提议将拟可微函数重命名为拟差凸函数，统一处理复合拟差凸规划的迭代凸规划下降算法，并建立其子序列、序列收敛性及收敛速率，推进了该非凸不可微优化领域的基础理论与应用。

A Note on Nesterov's Accelerated Method in Nonconvex Optimization: a Weak Estimate Sequence Approach

We present a variant of accelerated gradient descent algorithms, adapted from Nesterov's optimal first-order methods, for weakly-quasi-convex and weakly-quasi-strongly-convex functions. We show that by tweaking the so-called estimate sequence method, the derived algorithm achieves optimal convergence rate for weakly-quasi-convex and weakly-quasi-strongly-convex in terms of oracle complexity. In particular, for a weakly-quasi-convex function with Lipschitz continuous gradient, we require $O(\frac{1}{\sqrt{\varepsilon}})$ iterations to acquire an $\varepsilon$-solution; for weakly-quasi-strongly…

Micro Water-Energy Nexus: Optimal Demand-Side Management and Quasi-Convex Hull Relaxation

This paper investigates the water network's potential ability to provide demand response services to the power grid under the framework of a distribution-level water-energy nexus (micro-WEN). In particular, the hidden controllability of water loads, such as irrigation systems, was closely studied to improve the flexibility of electrical grids. A optimization model is developed for the demand-side management (DSM) of micro-WEN, and the simulation results assert that grid flexibility indeed benefits from controllable water loads. Although the proposed optimal DSM model is an intractable mixed-in…

Acceleration in First Order Quasi-strongly Convex Optimization by ODE Discretization

We study gradient-based optimization methods obtained by direct Runge-Kutta discretization of the ordinary differential equation (ODE) describing the movement of a heavy-ball under constant friction coefficient. When the function is high order smooth and strongly convex, we show that directly simulating the ODE with known numerical integrators achieve acceleration in a nontrivial neighborhood of the optimal solution. In particular, the neighborhood can grow larger as the condition number of the function increases. Furthermore, our results also hold for nonconvex but quasi-strongly convex objec…

From Convex Optimization to MDPs: A Review of First-Order, Second-Order and Quasi-Newton Methods for MDPs

In this paper we present a review of the connections between classical algorithms for solving Markov Decision Processes (MDPs) and classical gradient-based algorithms in convex optimization. Some of these connections date as far back as the 1980s, but they have gained momentum in recent years and have lead to faster algorithms for solving MDPs. In particular, two of the most popular methods for solving MDPs, Value Iteration and Policy Iteration, can be linked to first-order and second-order methods in convex optimization. In addition, recent results in quasi-Newton methods lead to novel algori…

Sufficient conditions for the value function and optimal strategy to be even and quasi-convex

Sufficient conditions are identified under which the value function and the optimal strategy of a Markov decision process (MDP) are even and quasi-convex in the state. The key idea behind these conditions is the following. First, sufficient conditions for the value function and optimal strategy to be even are identified. Next, it is shown that if the value function and optimal strategy are even, then one can construct a "folded MDP" defined only on the non-negative values of the state space. Then, the standard sufficient conditions for the value function and optimal strategy to be monotone are…

梯度法、共轭梯度法和拟牛顿法中不精确自适应步长的统一框架

A unified framework for inexact adaptive stepsizes in the gradient methods, the conjugate gradient methods and the quasi-Newton methods for strictly convex quadratic optimization

本文提出了一种统一框架，用于梯度法、共轭梯度法和拟牛顿法在严格凸二次优化中的不精确自适应步长，分析了其全局收敛性及收敛速率，并通过数值结果验证了其有效性。

A Variable Sample-size Stochastic Quasi-Newton Method for Smooth and Nonsmooth Stochastic Convex Optimization

Classical theory for quasi-Newton schemes has focused on smooth deterministic unconstrained optimization while recent forays into stochastic convex optimization have largely resided in smooth, unconstrained, and strongly convex regimes. Naturally, there is a compelling need to address nonsmoothness, the lack of strong convexity, and the presence of constraints. Accordingly, this paper presents a quasi-Newton framework that can process merely convex and possibly nonsmooth (but smoothable) stochastic convex problems. We propose a framework that combines iterative smoothing and regularization wit…

Incremental Subgradient Methods for Minimizing The Sum of Quasi-convex Functions

The sum of ratios problem has a variety of important applications in economics and management science, but it is difficult to globally solve this problem. In this paper, we consider the minimization problem of a sum of a number of nondifferentiable quasi-convex component functions over a closed and convex set, which includes the sum of ratios problem as a special case. The sum of quasi-convex component functions is not necessarily to be quasi-convex, and so, this study goes beyond quasi-convex optimization. Exploiting the structure of the sum-minimization problem, we propose a new incremental …

关于拟L$^{\ atural}$-凸函数最小化的最速下降算法的注记

Note on Steepest Descent Algorithm for Quasi L$^{\natural}$-convex Function Minimization

该研究定义了半严格拟L$^{\ atural}$-凸函数这类离散拟凸函数，证明L$^{\ atural}$-凸函数最小化的最速下降算法对其适用，并拓展迭代次数分析，揭示该算法的测地性质。

Quasi-Harmonic Constraints for Toric Bézier Surfaces

Toric Bézier patches generalize the classical tensor-product triangular and rectangular Bézier surfaces, extensively used in $CAGD$. The construction of toric Bézier surfaces corresponding to multi-sided convex hulls for known boundary mass-points with integer coordinates (in particular for trapezoidal and hexagonal convex hulls) is given. For these toric Bézier surfaces, we find approximate minimal surfaces obtained by extremizing the quasi-harmonic energy functional. We call these approximate minimal surfaces as the quasi-harmonic toric Bézier surfaces. This is achieved by imposing the vanis…

基于迭代加权方法的高效非凸拟范数球投影计算

Efficient Computation of the Non-convex Quasi-norm Ball Projection with Iterative Reweighted Approach

针对非凸非Lipschitz的 $\ell_p$ 拟范数球投影难题，提出基于局部近似构建凹替代函数的方法，通过构建更紧致子问题改进迭代加权算法，实现了全局收敛与卓越计算效率。

Simulated annealing with hit-and-run for convex optimization: rigorous complexity analysis and practical perspectives for copositive programming

We give a rigorous complexity analysis of the simulated annealing algorithm by Kalai and Vempala [Math of OR 31.2 (2006): 253-266] using the type of temperature update suggested by Abernethy and Hazan [arXiv 1507.02528v2, 2015]. The algorithm only assumes a membership oracle of the feasible set, and we prove that it returns a solution in polynomial time which is near-optimal with high probability. Moreover, we propose a number of modifications to improve the practical performance of this method, and present some numerical results for test problems from copositive programming.

面向一般凸函数的多割线拟牛顿方法的进展

Advancing Multi-Secant Quasi-Newton Methods for General Convex Functions

本文针对一般凸函数提出对称化且经PSD扰动的多割线拟牛顿方法，证明其超线性收敛并展示数值优势，同时研究其有限内存BFGS扩展在凸与非凸问题上的加速收敛效果。

拟凸的Rockafellar定理

The quasi-convex Rockafellar theorem

该研究建立了拟凸情形下的Rockafellar定理，证明可分Banach空间中循环拟单调算子包含于拟凸函数的法锥算子，推广了Afriat定理并得到最优运输问题的Kantorovich型势函数。

Beyond Convexity: Stochastic Quasi-Convex Optimization

Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD). The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. In this paper we analyze a stochastic version of NGD and prove its convergence to a global minimum for a wider class of functions: we require the functions to be quasi-convex and locally-Lipschitz. Quasi…

Chance-constrained quasi-convex optimization with application to data-driven switched systems control

We study quasi-convex optimization problems, where only a subset of the constraints can be sampled, and yet one would like a probabilistic guarantee on the obtained solution with respect to the initial (unknown) optimization problem. Even though our results are partly applicable to general quasi-convex problems, in this work we introduce and study a particular subclass, which we call "quasi-linear problems". We provide optimality conditions for these problems. Thriving on this, we extend the approach of chance-constrained convex optimization to quasi-linear optimization problems. Finally, we s…

Disciplined Quasiconvex Programming

We present a composition rule involving quasiconvex functions that generalizes the classical composition rule for convex functions. This rule complements well-known rules for the curvature of quasiconvex functions under increasing functions and pointwise maximums. We refer to the class of optimization problems generated by these rules, along with a base set of quasiconvex and quasiconcave functions, as disciplined quasiconvex programs. Disciplined quasiconvex programming generalizes disciplined convex programming, the class of optimization problems targeted by most modern domain-specific langu…

Convex Optimization over Fixed Value Point Set of Quasi-Nonexpansive Random Operators on Hilbert Spaces

In this paper, a new optimization framework is defined that includes the optimization framework recently proposed in [1]-[2] as a special case. The convex optimization in [1]-[2] includes centralized optimization and distributed optimization over random networks, so does the optimization defined here. It is shown that the proposed algorithm in [2] converges almost surely and in mean square to a solution of the optimization problem here under suitable assumptions.

Uniform quasi-convex optimisation via Extremum Seeking

The paper deals with a well-known extremum seeking scheme by proving uniformity properties with respect to the amplitudes of the dither signal and of the cost function. Those properties are then used to show that the scheme guarantees the global minimiser to be semi-global practically stable despite the presence of local saddle points. To achieve these results, we analyse the average system associated with the extremum seeking scheme via arguments based on the Fourier series.

Optimization Of Quasi-convex Function Over Product Measure Sets

We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures sets, that are non necessarily compact but generated by their extreme points. We show that the maximum of a quasi-convex lower semicontinuous function on this product space is reached on the tensorial product of finite mixtures of extreme points. Our work is an extension of the Bauer maximum principle in three different aspects. First, we only assume that the objective functional is quasi-convex. Secondly, the optimization is performed over a space built as a product of measures sets…

准Hankel低秩矩阵补全：一种凸松弛

Quasi-Hankel low-rank matrix completion: a convex relaxation

本文研究了结构化矩阵补全问题，特别是Hankel和准Hankel矩阵补全，扩展了现有结果至复数Hankel和准Hankel矩阵的低秩补全。

近似拟牛顿方法在正则化凸优化中的应用：线性和加速次线性收敛率

Proximal Quasi-Newton Methods for Regularized Convex Optimization with Linear and Accelerated Sublinear Convergence Rates

本文研究了正则化凸优化中近似拟牛顿方法的收敛性，分析了强凸情况下精确与不精确设置的收敛性质，并探讨了加速变体的实用性与性能。