Mootta Prangprakhon, Nimit Nimana, Narin Petrot
We consider the variational inequality problem over the intersection of fixed point sets of firmly nonexpansive operators. In order to solve the problem, we present an algorithm and subsequently show the strong convergence of the generated sequence to the solution of the considered problem.
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An algorithm is developed to optimize vehicle speed trajectory over multiple signalized intersections with known traffic signal information to minimize fuel consumption and travel time, and to meet ride comfort requirements using sequential convex optimization method. A comparison between the proposed method and dynamic programming is carried out to verify its optimality. In addition, vehicle motion during turning is studied because of its significant effect on fuel consumption and travel time.
This paper deals with the solving of variational inequality problem where the constrained set is given as the intersection of a number of fixed-point sets. To this end, we present an extrapolated sequential constraint method. At each iteration, the proposed method is updated based on the ideas of a hybrid conjugate gradient method used to accelerate the well-known hybrid steepest descent method, and an extrapolated cyclic cutter method for solving a common fixed point problem. We prove strong convergence of the method under some suitable assumptions of step-size sequences. We finally show the …
A Sequential Quadratic Programming Method for Optimization with Stochastic Objective Functions, Deterministic Inequality Constraints and Robust Subproblems
本文将鲁棒序列二次规划方法推广到随机目标函数与确定性约束问题，结合随机线搜索全局化步长，并证明算法几乎必然收敛到KKT点。
A sequential quadratic optimization algorithm for minimizing an objective function defined by an expectation subject to nonlinear inequality and equality constraints is proposed, analyzed, and tested. The context of interest is when it is tractable to evaluate constraint function and derivative values in each iteration, but it is intractable to evaluate the objective function or its derivatives in any iteration, and instead an algorithm can only make use of stochastic objective gradient estimates. Under loose assumptions, including that the gradient estimates are unbiased, the algorithm is pro…
A smoothing extended sequential quadratic method for difference-of-convex optimization over a convex composite inequality constraint
提出一种平滑扩展序列二次方法求解差凸目标与凸复合不等式约束问题，通过平滑近似和参数更新实现O(ε^{-3})迭代复杂度，并证明凸情形下序列收敛及局部收敛率。
Sequential Extremal Principle and Necessary Conditions for Minimizing Sequences
本文将极值性定义从固定点推广到点列，在Fréchet和Clarke法锥框架下建立序列极值原理，并给出约束优化中极小化序列的序列必要条件。
In this paper, we studied the equilibrium problem where the bi-function may be quasiconvex with respect to the second variable and the feasible set is the intersection of a finite number of convex sets. We propose a projection-algorithm, where the projection can be computed independently onto each component set. The convergence of the algorithm is investigated and numerical examples for a variational inequality problem involving affine fractional operator are provided to demonstrate the behavior of the algorithm.
We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. In particular, we give simple formulas for split cuts for essentially all convex sets…
Fast projection onto the intersection of simplex and singly linear constraint and its generalized Jacobian
本文针对单纯形与单个线性不等式约束交集上的投影问题，提出结合 Lagrange 对偶与改进割线法的 LRSA，并推导其广义 HS-Jacobian，实验表明其优于 SSN 与 Gurobi。
In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic p…
We present a framework to obtain valid inequalities for a reverse convex set: the set of points in a polyhedron that lie outside a given open convex set. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. An intersection cut is a well-known valid inequality for a reverse convex set that is generated from a basic solution that lies within the convex set. We introduce a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We first propose an extension to intersection cuts that…
On circumcentered direct methods for monotone variational inequality problems
本文提出两种基于圆心加速的直接方法，用于高效求解涉及参数单调和单调算子的变分不等式问题，实验表明其在计算效率上有显著优势。
We study nonlinear optimization problems with a stochastic objective and deterministic equality and inequality constraints, which emerge in numerous applications including finance, manufacturing, power systems and, recently, deep neural networks. We propose an active-set stochastic sequential quadratic programming (StoSQP) algorithm that utilizes a differentiable exact augmented Lagrangian as the merit function. The algorithm adaptively selects the penalty parameters of the augmented Lagrangian and performs a stochastic line search to decide the stepsize. The global convergence is established:…
This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs (QPs) and approximate projections arising from partially solving the QPs are sufficient for multiple-term superlinear convergence for nonsmooth problems. Second, we identify additional regularity, which we call the second order supporting hyperplane property (SOSH), that gives multiple-term quadratic convergence. Third, we show that these fast convergence res…
A Noise Tolerant SQP Algorithm for Inequality Constrained Optimization
本文提出了一种鲁棒的序列二次规划算法，用于处理函数和导数评估中的有界噪声，通过松弛策略实现噪声容忍，并通过数值实验验证了算法在噪声下的收敛性。
In this article, a globally convergent sequential quadratic programming (SQP) method is developed for multi-objective optimization problems with inequality type constraints. A feasible descent direction is obtained using a linear approximation of all objective functions as well as constraint functions. The sub-problem at every iteration of the sequence has feasible solution. A non-differentiable penalty function is used to deal with constraint violations. A descent sequence is generated which converges to a critical point under the Mangasarian-Fromovitz constraint qualification along with some…
A Sequential Descent Method for Global Optimization
提出一种序列下降方法，通过重复下降梯度搜索并利用交点集合选择最优初始点，逐步改进局部最小值，最终找到全局最小值，该方法具有单调收敛性，并在非凸多维问题上验证了有效性。
Randomized Feasibility-Update Algorithms for Stochastic Variational Inequality Problems
本文针对可行域为多个凸水平集交集的随机单调变分不等式，提出基于随机可行性更新的修正Korpelevich和Popov方法，引入修正对偶间隙函数并证明收敛速率，仿真验证了其有效性。
This paper presents a convex sufficient condition for solving a system of nonlinear equations under parametric changes and proposes a sequential convex optimization method for solving robust optimization problems with nonlinear equality constraints. By bounding the nonlinearity with concave envelopes and using Brouwer's fixed point theorem, the sufficient condition is expressed in terms of closed-form convex inequality constraints. We extend the result to provide a convex sufficient condition for feasibility under bounded uncertainty. Using these conditions, a non-convex optimization problem c…
Randomized Feasibility-Update Algorithms for Variational Inequality Problems
针对多智能体博弈衍生的高计算成本变分不等式问题，提出三类基于随机可行性更新的求解算法，证明其几乎必然收敛性并给出收敛速率，经双智能体博弈仿真验证了性能。
Simultaneous perturbation stochastic approximation (SPSA) is widely used in stochastic optimization due to its high efficiency, asymptotic stability, and reduced number of required loss function measurements. However, the standard SPSA algorithm needs to be modified to deal with constrained problems. In recent years, sequential quadratic programming (SQP)-based projection ideas and penalty ideas have been analyzed. Both ideas have convergence results and a potentially wide range of applications, but with some limitations in practical consideration, such as computation time, complexity, and fea…
A computationally efficient method to solve non-convex programming problems with linear equality constraints is presented. The proposed method is based on a recursively feasible and descending sequential convex programming procedure proven to converge to a locally optimal solution. Assuming that the first convex problem in the sequence is feasible, these properties are obtained by convexifying the non-convex cost and inequality constraints with inner-convex approximations. Additionally, a computationally efficient method is introduced to obtain inner-convex approximations based on Taylor serie…
In this paper, we consider a nonlinear semi-infinite program that minimizes a function including a log-determinant (logdet) function over positive definite matrix constraints and infinitely many convex inequality constraints, called SIPLOG for short. The main purpose of the paper is to develop an algorithm for computing a Karush-Kuhn-Tucker (KKT) point for the SIPLOG efficiently. More specifically, we propose an interior point sequential quadratic programming-type method that inexactly solves a sequence of semi-infinite quadratic programs approximating the SIPLOG. Furthermore, to generate a se…
Feedback Stackelberg-Nash equilibria in difference games with quasi-hierarchical interactions and inequality constraints
本文针对具有耦合不等式约束的差分博弈，提出反馈Stackelberg-Nash均衡概念，通过动态规划递归求解，并在线性二次情形下转化为线性互补问题，最后用双寡头博弈验证。
Optimization-based Coordination of Traffic Lights and Automated Vehicles at Intersections
本文将信号化交叉口交通信号灯与自动驾驶车辆协调建模为混合整数非线性规划问题，提出上下层分解方法，利用下层解和参数优化技术通过序列二次规划求解，数值算例验证了其有效性。
Splitting Guarantees for Prophet Inequalities via Nonlinear Systems
针对i.i.d. k选择先知不等式问题，引入推广Hill-Kertz方程的非线性微分方程组，通过无限维线性规划与对偶拟合得渐近近似比下界，还为随机序贯分配问题建立紧近似比。
A Trust-Region Interior-Point Stochastic Sequential Quadratic Programming Method
本文提出了一种针对随机目标函数和确定性非线性约束的优化方法，结合信任区域内点和随机序列二次规划技术，实现全局收敛性保证。
We discuss an online decentralized decision making problem where the agents are coupled with affine inequality constraints. Alternating Direction Method of Multipliers (ADMM) is used as the computation engine and we discuss the convergence of the algorithm in an online setting. To be specific, when decisions have to be made sequentially with a fixed time step, there might not be enough time for the ADMM to converge before the scenario changes and the decision needs to be updated. In this case, a suboptimal solution is employed and we analyze the optimality gap given the convergence condition. …
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