Hasan Yilmaz
We provide a generalization of first-order necessary conditions of optimality for infinite-dimensional optimization problems with a finite number of inequality constraints and with a finite number of inequality and equality constraints. Our assumptions on the differentiability of the functions are weaker than those of existing results.
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The concept of uniform tangent sets was introduced and discussed in [3 - Krastanov, Ribarska, SIAM J. Control Optim., 55(3), 2017]. This study is devoted to their further investigation and to generalization of the abstract Lagrange multiplier rule from [3].
In this note, three Lagrange multiplier rules introduced in the literature for set valued optimization problems are compared. A generalization of all three results is given which proves that under rather mild assumptions, $x$ is a weak solution to the constrained problem, if and only if it is a weak solution to the Lagrangian.
We use an extension to the infinite dimension of the rank theorem of the differential calculus to establish a Karush-Huhn-Tucker theorem for optimization problems in Banach spaces. We provide an application to variational problems on bounded processus under equality constraints.
We establish new results of first-order necessary conditions of optimality for finite-dimensional problems with inequality constraints and for problems with equality and inequality constraints, in the form of John's theorem and in the form of Karush-Kuhn-Tucker's theorem. In comparison with existing results we weaken assumptions of continuity and of differentiability.
Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings: decoupling approach revisited
本文在非Lipschitzian设定下再探讨了解耦方法，提出了拟一致下确界、下半连续性和极小值的较弱概念，建立了刻画两函数之和拟一致$\varepsilon$-极小值的原始与对偶必要条件及模糊乘子规则，并应用于稀疏最优控制。
Multiplier rules for Dini-derivatives in a topological vector space
本文在拓扑向量空间和Banach空间中，将最优性条件从Fréchet、Gateaux可微及Clarke次微分情形推广到Dini可微函数，并扩展了Pourciau的凸性结果至Dini-伪凸性。
Infinite Dimensional Multipliers and Pontryagin Principles for Discrete-Time Problems
本文改进了无限时间框架下无限维状态和控制空间的庞特里亚金原理，通过有限时间缩减方法和泛函分析引理实现目标。
The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems, and show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. Fo…
On Lagrange multipliers of constrained optimization in Hilbert spaces
本文提出了一种新的分解框架，用于研究Hilbert空间中约束优化问题的拉格朗日乘数，揭示了有限维与无限维空间中拉格朗日乘数理论的本质差异，并建立了二次规划方法的数学基础。
This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-SOS relaxations are used to solve the polynomial optimizattion relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimizatio…
This paper proposes a partially inexact alternating direction method of multipliers for computing approximate solution of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic interation-complexity bounds for the proposed method are established. To the best of our know…
In this paper, we accomplish a unified convergence analysis of a second-order method of multipliers (i.e., a second-order augmented Lagrangian method) for solving the conventional nonlinear conic optimization problems.Specifically, the algorithm that we investigated incorporates a specially designed nonsmooth (generalized) Newton step to furnish a second-order update rule for the multipliers.We first show in a unified fashion that under a few abstract assumptions, the proposed method is locally convergent and possesses a (nonasymptotic) superlinear convergence rate, even though the penalty par…
In this paper we introduce the notions of critical and noncritical multipliers for subdifferential variational systems extending to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush-Kuhn-Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax prob…
We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse optimal control problem (which is an instance of an infinite-dimensional bilevel optimization problem). The multipliers for the M-stationarity system can be constructed via convex combinations of various multipliers to auxiliary, linear problems. However, proving the existence of the multipliers to these auxiliary problems is difficult and only possible in s…
An inertial ADMM for a class of nonconvex composite optimization with nonlinear coupling constraints
本文针对一类具有非线性耦合约束的非凸多块优化问题，提出一种结合惯性技术和带松弛参数的非标准乘子更新规则的交替方向乘子法，并证明了算法的子序列收敛性与全局收敛性。
The alternating direction method of multipliers (ADMM) is a widely used method for solving many convex minimization models arising in signal and image processing. In this paper, we propose an inertial ADMM for solving a two-block separable convex minimization problem with linear equality constraints. This algorithm is obtained by making use of the inertial Douglas-Rachford splitting algorithm to the corresponding dual of the primal problem. We study the convergence analysis of the proposed algorithm in infinite-dimensional Hilbert spaces. Furthermore, we apply the proposed algorithm on the rob…
Mangasarian-Fromovitz-type constraint qualification and optimality conditions for smooth infinite programming problems
本文针对光滑无限规划问题提出推广的Mangasarian-Fromovitz型约束规范GPMFCQ，并借助Hurwicz集或非线性Farkas-Minkowski条件证明拉格朗日乘子的存在性。
In this paper we develop and analyse convex searches for Zames--Falb multipliers. We present two different approaches: Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) multipliers. The set of FIR multipliers is complete in that any IIR multipliers can be phase-substituted by an arbitrarily large order FIR multiplier. We show that searches in discrete-time for FIR multipliers are effective even for large orders. As expected, the numerical results provide the best $\ell_{2}$-stability results in the literature for slope-restricted nonlinearities. Finally, we demonstrate that the…
The main objective of this work is to study the existence of Lagrange multipliers for infinite dimensional problems under Gâteux differentiability assumptions on the data. Our investigation follows two main steps: the proof of the existence of Lagrange multipliers under a calmness assumption on the constraints and the study of sufficient conditions, which only use the Gâteaux derivative of the function defining the constraint, that ensure this assumption.
Lagrange Multipliers in locally convex spaces
本文研究了Hausdorff局部凸空间中含无穷多约束的数学规划问题，提出一般拉格朗日乘子规则，用Gateaux可微性替代Fréchet可微性，并引入“容许集”替代闭凸锥以推广现有结果。
The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for large-scale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking. In this paper we find the optimal algorithm parameters that minimize the convergence factor of the ADMM iterates in the context of l2-regularized minimization and constrained quadratic programming. Numerical examples show that our parameter selection rules significantly…
Strong duality in infinite convex optimization
本文针对无限维凸优化提出两种含无穷多对偶变量与无穷和函数的拉格朗日型对偶形式，在 Slater 条件下实现零对偶间隙与可解性，并利用无穷和次微分法则导出一般最优性条件。
Polynomial Optimization Relaxations for Generalized Semi-Infinite Programs
本文针对多项式形式的广义半无限规划，提出基于拉格朗日乘子表达式和多项式扩张的多项式优化松弛层级方法，证明其收敛性，并给出凸无穷约束情形下的精确求解条件，数值实验验证了计算效率。
Non-stationary Douglas-Rachford and alternating direction method of multipliers: adaptive stepsizes and convergence
本文重新审视经典Douglas-Rachford方法用于求解两个最大单调算子之和的零点问题，提出了一种自适应步长规则，通过分析线性情况并开发出无需调整步长的策略，证明了非平稳DR方案在可和增量步长序列下的收敛性，并由此推导出相关的非平稳交替方向乘数法（ADMM），并通过数值实验验证了所提方法的效率。
Alternating Direction Method of Multipliers (ADMM) algorithm has been widely adopted for solving the distributed optimization problem (DOP). In this paper, a new distributed parallel ADMM algorithm is proposed, which allows the agents to update their local states and dual variables in a completely distributed and parallel manner by modifying the existing distributed sequential ADMM. Moreover, the updating rules and storage method for variables are illustrated. It is shown that all the agents can reach a consensus by asymptotically converging to the optimal solution. Besides, the global cost fu…
There has been a recent surge of interest in physics-based solvers for combinatorial optimization problems. We present a dynamical solver for the Ising problem that is comprised of a network of coupled parametric oscillators and show that it implements Lagrange multiplier constrained optimization. We show that the pump depletion effect, which is intrinsic to parametric oscillators, enforces binary constraints and enables the system's continuous analog variables to converge to the optimal binary solutions to the optimization problem. Moreover, there is an exact correspondence between the equati…
Process-Based Lagrange Multipliers for Nonconvex Set-Valued Optimization
本文提出了一种基于闭合凸过程的非凸集值优化拉格朗日乘子理论，通过几何框架扩展分离原理，并展示了其在无限维情况下的应用。
This paper investigates the necessary optimality conditions for uniformly overtaking optimal control on infinite horizon in the free end case. %with free right endpoint. In the papers of S.M.Aseev, A.V.Kryazhimskii, V.M.Veliov, K.O.Besov there was suggested the boundary condition for equations of the Pontryagin Maximum Principle. Each optimal process corresponds to a unique solution satisfying the boundary condition. Following A.Seierstad's idea, in this paper we prove a more general geometric variety of that boundary condition. We show that this condition is necessary for uniformly overtaking…
We establish differentiability properties of the value function of problems of Static Optimization in an abstract infinite dimensional setting and we apply that to problems of Calculus of Variations. We lighten the assumptions of existing results, notably by using G{â}teaux and Hadamard differentials. Moreover we use recently established Multipliers Rules.
Quantization of the parameters of machine learning models, such as deep neural networks, requires solving constrained optimization problems, where the constraint set is formed by the Cartesian product of many simple discrete sets. For such optimization problems, we study the performance of the Alternating Direction Method of Multipliers for Quantization ($\texttt{ADMM-Q}$) algorithm, which is a variant of the widely-used ADMM method applied to our discrete optimization problem. We establish the convergence of the iterates of $\texttt{ADMM-Q}$ to certain $\textit{stationary points}$. To the bes…
This paper contains selected applications of the new tangential extremal principles and related results developed in Part I to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraints
Integral Quadratic Constraints with Infinite-Dimensional Channels
本文提出一个基于积分二次约束（IQC）的框架，用于偏微分方程和时滞微分方程的输入-输出稳定性分析，通过无限维乘子和PIE表示及KYP引理实现，并用四个示例验证。
Learning solutions to some toy constrained optimization problems in infinite dimensional Hilbert spaces
本文提出了罚函数法和增广Lagrangian法在无限维Hilbert空间中的深度学习实现，并在变分法与物理学的玩具问题上验证了其近似效果，同时利用Lagrange乘子更新规则实现了显著加速。
We study several versions of the alternating direction method of multipliers (ADMM) for solving the convex problem of finding the distance between two ellipsoids and the nonconvex problem of finding the distance between the boundaries of two ellipsoids. In the convex case we present the ADMM with and without automatic penalty updates and demonstrate via numerical experiments on problems of various dimensions that our methods significantly outperform all other existing methods for finding the distance between ellipsoids. In the nonconvex case we propose a heuristic rule for updating the penalty…
In this paper, we explore aggregative games over networks of multi-integrator agents with coupled constraints. To reach the general Nash equilibrium of an aggregative game, a distributed strategy-updating rule is proposed by a combination of the coordination of Lagrange multipliers and the estimation of the aggregator. Each player has only access to partial-decision information and communicates with his neighbors in a weight-balanced digraph which characterizes players' preferences as to the values of information received from neighbors. We first consider networks of double-integrator agents a…
The canonical solution methodology for finite constrained Markov decision processes (CMDPs), where the objective is to maximize the expected infinite-horizon discounted rewards subject to the expected infinite-horizon discounted costs constraints, is based on convex linear programming. In this brief, we first prove that the optimization objective in the dual linear program of a finite CMDP is a piece-wise linear convex function (PWLC) with respect to the Lagrange penalty multipliers. Next, we propose a novel two-level Gradient-Aware Search (GAS) algorithm which exploits the PWLC structure to f…
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