In this paper, we propose a two-phase algorithm for solving continuous rank-one quadratic knapsack problems (R1QKP). In particular, we study the solution structure of the problem without the knapsack constraint. We propose an $O(n\log n)$ algorithm in this case. We then use the solution structure to propose an $O(n^2\log n)$ algorithm that finds an interval containing the optimal value of the Lagrangian dual of R1QKP. In the second phase, we solve the restricted Lagrangian dual problem using a traditional single-variable optimization method. We perform a computational test on random instances and compare our algorithm with the general solver CPLEX.
In this paper, we propose a robust optimization-based heuristic algorithm for the chance-constrained binary knapsack problem (CKP). We assume that the weights of items are independent normally distributed. By utilizing the properties of the submodular function, the proposed method approximates the CKP to the robust knapsack problem with a cardinality constrained uncertainty set parameterized by a uncertainty budget parameter. The proposed approach obtains a heuristic solution by solving the approximated robust knapsack problem whose optimal solution can be obtained by solving the ordinary bina…
QMKPy provides a Python framework for modeling and solving the quadratic multiple knapsack problem (QMKP). It is primarily aimed at researchers who develop new solution algorithms for the QMKP. QMKPy therefore mostly functions as a testbed to quickly implement novel algorithms and compare their results with existing ones. However, the package also already includes implementations of established algorithms for those who only need to solve a QMKP as part of their research.
A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-Hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by a be…
In this paper, we consider an SDP relaxation of the quadratic knapsack problem (QKP). After using the Burer-Monteiro factorization, we get a non-convex optimization problem, whose feasible region is an algebraic variety. Although there might be non-regular points on the algebraic variety, we prove that the algebraic variety is a smooth manifold except for a trivial point for a generic input data. We also analyze the local geometric properties of non-regular points on this algebraic variety. In order to maintain the equivalence between the SDP problem and its non-convex formulation, we derive a…
Dynamic-Threshold Algorithms for the Continuous Quadratic Knapsack Problem: Reset Mechanisms and Complexity
Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University（福建省应用数学中心，福州大学数学与统计学院）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
We propose a method for finding approximate solutions to multiple-choice knapsack problems. To this aim we transform the multiple-choice knapsack problem into a bi-objective optimization problem whose solution set contains solutions of the original multiple-choice knapsack problem. The method relies on solving a series of suitably defined linearly scalarized bi-objective problems. The novelty which makes the method attractive from the computational point of view is that we are able to solve explicitly those linearly scalarized bi-objective problems with the help of the closed-form formulae. Th…
We consider a parametric convex quadratic programming, CQP, relaxation for the quadratic knapsack problem, QKP. This relaxation maintains partial quadratic information from the original QKP by perturbing the objective function to obtain a concave quadratic term. The nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to the matrix space. We present a primal-dual interior point method to optimize the perturbation of the quadratic function, in a search for the tightest upper bound for the \QKPp. We prove that the same perturbation approa…
A scenario-cluster-based and enhanced progressive hedging algorithm for the two-stage stochastic quadratic knapsack problem
GERAD, CIRRELT & Department of Analytics, Operations and Information Technology, Université du Québec à Montréal（蒙特利尔大学 GERAD、CIRRELT 与分析、运营与信息技术系）；GERAD, CIRRELT & Department of Operations and Decision Systems, Université Laval（魁北克大学拉瓦尔分校 GERAD、CIRRELT 与运营与决策系统系）；GERAD, CRM & Department of Analytics, Operations and Information Technology, Université du Québec à Montréal（蒙特利尔大学 GERAD、CRM 与分析、运营与信息技术系）
We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set. We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations…
University of Alberta（阿尔伯塔大学）；George Mason University（乔治梅森大学）；Texas A&M University-Commerce（德州农工大学康默斯分校）；Sandia National Laboratories（桑迪亚国家实验室）；New York University（纽约大学）
0-1 knapsack is of fundamental importance in computer science, business, operations research, etc. In this paper, we present a deep learning technique-based method to solve large-scale 0-1 knapsack problems where the number of products (items) is large and/or the values of products are not necessarily predetermined but decided by an external value assignment function during the optimization process. Our solution is greatly inspired by the method of Lagrange multiplier and some recent adoptions of game theory to deep learning. After formally defining our proposed method based on them, we develo…
In this paper, we consider the classic stochastic (dynamic) knapsack problem, a fundamental mathematical model in revenue management, with general time-varying random demand. Our main goal is to study the optimal policies, which can be obtained by solving the dynamic programming formulated for the problem, both qualitatively and quantitatively. It is well-known that when the demand size is fixed and the demand distribution is stationary over time, the value function of the dynamic programming exhibits extremely useful first and second order monotonicity properties, which lead to monotonicity p…
This paper studies the multi-item newsvendor problem with a constrained budget and information about demand limited to its range, mean and mean absolute deviation. We consider a minimax model that determines order quantities by minimizing the expected overage and underage costs for the worst-case demand distributions. The resulting optimization problem turns out to be solvable by a method reminiscent of the greedy algorithm that solves the continuous knapsack problem, purchasing items in order of marginal value. This method has lower computational complexity compared to directly solving the mo…
Dynamic vehicle routing problems (DVRPs) arise in several applications such as technician routing, meal delivery, and parcel shipping. We consider the DVRP with stochastic customer requests (DVRPSR), in which vehicles must be routed dynamically with the goal of maximizing the number of served requests. We model the DVRPSR as a multi-stage optimization problem, where the first-stage decision defines route plans for serving scheduled requests. Our main contributions are knapsack-based linear models to approximate accurately the expected reward-to-go, measured as the number of accepted requests, …
We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combina…
We introduce a new technique for solving uni-parametric versions of linear programs, convex quadratic programs, and linear complementarity problems in which a single parameter is permitted to be present in any of the input data. We demonstrate the use of our method on a small, motivating example and present the results of a small number of computational tests demonstrating its utility for larger scale problems.
In this work we consider the problem of computing the $(\min, +)$-convolution of two sequences $a$ and $b$ of lengths $n$ and $m$, respectively, where $n \geq m$. We assume that $a$ is arbitrary, but $b_i = f(i)$, where $f(x) \colon [0,m) \to \mathbb{R}$ is a function with one of the following properties: 1. the linear case, when $f(x) =β+ α\cdot x$; 2. the monotone case, when $f(i+1) \geq f(i)$, for any $i$; 3. the convex case, when $f(i+1) - f(i) \geq f(i) - f(i-1)$, for any $i$; 4. the concave case, when $f(i+1) - f(i) \leq f(i) - f(i-1)$, for any $i$; 5. the piece-wise linear case, when $f…
This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form $f + h$ where $f$ is a differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of…
This paper demonstrates a mathematically correct and computationally powerful method for solving 3D topology optimization problems. This method is based on canonical duality theory (CDT) developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard knapsack problem in topology optimization can be solved deterministically in polynomial time via a canonical penalty-duality (CPD) method to obtain precise 0-1 global optimal solution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the…
Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This work focuses on the subclass of quadratic programs with linear complementarity constraints. A novel approach to solving a penalty reformulation using sequential convex programming and a homotopy on the penalty parameter is introduced. Linearizing the necessarily nonconvex penalty function yields convex quadratic subproblems, which have a constant Hessian matrix throughout all iterates. This allows solution comput…
A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost functional and a surjectivity condition on the linear constraint mapping. Based on the Lagrangian duality, two approaches are proposed to solve the constrained stochastic LQ problem. First, a theoretical method is given to construct the closed-form solution by the strong duality. Second, an iterative algorithm, called augmented Lagrangian method (ALM), is proposed…
In this paper, we discuss the solution of a Quadratic Eigenvalue Complementarity Problem (QEiCP) by using Difference of Convex (DC) programming approaches. We first show that QEiCP can be represented as dc programming problem. Then we investigate different dc programming formulations of QEiCP and discuss their dc algorithms based on a well-known method -- DCA. A new local dc decomposition is proposed which aims at constructing a better dc decomposition regarding to the specific feature of the target problem in some neighborhoods of the iterates. This new procedure yields faster convergence and…
Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause inconveniences when solutions are used for rigorous reasoning. We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision. It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors. Second, we provide an ef…
We propose a new formulation of quadratic optimization problems. The objective function $F(f(x),g(x))$ is given as composition of a quadratic function $F(z)$ with two $n$-variate quadratic functions $z_1=f(x)$ and $z_2=g(x).$ In addition, it incorporates with a set of linear inequality constraints in $z=(z_1,z_2)^T,$ while having an implicit constraint that $z$ belongs to the joint numerical range of $(f,g).$ The formulation is very general in the sense that it covers quadratic programming with a single quadratic constraint of all types, including the inequality-type, the equality-type, and th…
