Primal Feasibility Research Articles

The weighted Euclidean one-center problem in $${\mathbb {R}}^n$$

AbstractGiven a finite set of distinct points in $${\mathbb {R}}^n$$ R n and a positive weight for each point, primal and dual algorithms are developed for finding the Euclidean ball of minimum radius so that the weighted Euclidean distance between the center of the ball and each point is less than or equal to the radius of the ball. Each algorithm is based on a directional search method in which the search path at each iteration is either a ray or a two-dimensional circular arc in $${\mathbb {R}}^n$$ R n . At each iteration, a search path is constructed by intersecting bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional spheres. Each search path preserves complementary slackness and primal (dual) feasibility for the primal (dual) algorithm. The step size along each search path is determined explicitly. Test problems up to 1000 dimensions and 10,000 points were solved to optimality by both primal and dual algorithms. Computational results also show that these algorithms outperform several open-source SOCP codes.

Generating linear programming instances with controllable rank and condition number

Instance generation is crucial for linear programming algorithms, especially for the evaluation and verification of corresponding methods. This study proposes a general framework for designing linear programming instances based on the preset optimal solution. First, we give a constraint matrix generation method with controllable condition number and rank from the perspective of matrix decomposition. Based on the preset optimal solution, a bounded feasible linear programming instance is generated with the right-hand side and objective coefficients satisfying the primal feasibility and dual feasibility. In addition, we provide three neighborhood exchange operators and prove that instances generated under this method can fill the entire space of feasible bounded linear programming instances. We experimentally validate that the proposed schedule generates more controllable linear programming instances in rank and condition number, while neighborhood exchange operators construct more complex instances.

On the Primal Feasibility in Dual Decomposition Methods Under Additive and Bounded Errors

With the unprecedented growth of signal processing and machine learning application domains, there has been a tremendous expansion of interest in distributed optimization methods to cope with the underlying large-scale problems. Nonetheless, inevitable system-specific challenges such as limited computational power, limited communication, latency requirements, measurement errors, and noises in wireless channels impose restrictions on the exactness of the underlying algorithms. Such restrictions have appealed to the exploration of algorithms' convergence behaviors under inexact settings. Despite the extensive research conducted in the area, it seems that the analysis of convergences of dual decomposition methods concerning primal optimality violations, together with dual optimality violations is less investigated. Here, we provide a systematic exposition of the convergence of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">feasible points</i> in dual decomposition methods under inexact settings, for an important class of global consensus optimization problems. Convergences and the rate of convergences of the algorithms are mathematically substantiated, not only from a dual-domain standpoint but also from a primal-domain standpoint. Analytical results show that the algorithms converge to a neighborhood of optimality, the size of which depends on the level of underlying distortions.

A new randomized primal-dual algorithm for convex optimization with fast last iterate convergence rates

We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves and convergence rates in two cases: merely convexity and strong convexity, respectively, where k is the iteration counter and n is the number of block-coordinates. These rates are known to be optimal (up to a constant factor) when n = 1. Our convergence rates are obtained through three criteria: primal objective residual and primal feasibility violation, dual objective residual, and primal-dual expected gap. Moreover, our rates for the primal problem are on the last-iterate sequence. Our dual convergence guarantee requires additionally a Lipschitz continuity assumption. We specify our algorithm to handle two important special cases, where our rates are still applied. Finally, we verify our algorithm on two well-studied numerical examples and compare it with two existing methods. Our results show that the proposed method has encouraging performance on different experiments.

Multiple discrete-continuous choice models with additively separable utility functions and linear utility on outside good: Model properties and characterization of demand functions

In this paper, we enhance the current understanding of the properties of multiple discrete-continuous (MDC) choice models with additively separable (AS), independent and identically distributed (IID) utility functions, and linear utility form on the essential outside good. First, we highlight that the prior implementations of this model in the literature ignore primal feasibility conditions related to the budget constraint and the essential nature of outside good in formulating the model likelihood function. Second, we evaluate the suitability and performance of the model for alternative consumption patterns relative to the budget. In addition, we provide a systematic comparison of the performance of MDC choice models with the linear outside good utility form (i.e., the Lγ-profile model) and those with the non-linear outside good utility form (i.e., the NLγ-profile model). Third, for the Lγ-profile model with infinite budgets (i.e., when a very small proportion of the budget is allocated to inside goods), we derive the distributions of the resulting optimal demand functions and analytic expressions for the corresponding first and second moments, and identify a property that makes it easy to estimate the utility function parameters of an inside alternative even when consumption data is not available for other alternatives. In addition, perhaps for the first time in the literature, we show how an independent system of Tobit models can be derived as a restricted version of the utility-theoretic Lγ-profile MDC model structure. Finally, we apply the model for an empirical analysis of expenditure patterns of leisure trips from a domestic tourism survey sample of households in India.

Invariant Geometric Curvilinear Optimization with Restricted Evolution Dynamics

This paper begins with a geometric statement of constraint optimization problems, which include both equality and inequality-type restrictions. The cost to optimize is a curvilinear functional defined by a given differential one-form, while the optimal state to be determined is a differential curve connecting two given points, among all the curves satisfying some given primal feasibility conditions. The resulting outcome is an invariant curvilinear Fritz–John maximum principle. Afterward, this result is approached by means of parametric equations. The classical single-time Pontryagin maximum principle for curvilinear cost functionals is revealed as a consequence.

A Distributed Active Set Method for Model Predictive Control

Abstract This paper presents a novel distributed active set method for model predictive control of linear systems. The method combines a primal active set strategy with a decentralized conjugate gradient method to solve convex quadratic programs. An advantage of the proposed method compared to existing distributed model predictive algorithms is the primal feasibility of the iterates. Numerical results show that the proposed method can compete with the alternating direction method of multipliers in terms of communication requirements for a chain of masses example.

A Simpler and Faster Strongly Polynomial Algorithm for Generalized Flow Maximization

We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [34]. For the uncapacitated problem formulation, the complexity bound O ( mn ( m + n log n )log ( n 2 / m )) improves on the previous estimate by almost a factor O ( n 2 ). Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.

On the behavior of Lagrange multipliers in convex and nonconvex infeasible interior point methods

We analyze sequences generated by interior point methods (IPMs) in convex and nonconvex settings. We prove that moving the primal feasibility at the same rate as the barrier parameter $\mu$ ensures the Lagrange multiplier sequence remains bounded, provided the limit point of the primal sequence has a Lagrange multiplier. This result does not require constraint qualifications. We also guarantee the IPM finds a solution satisfying strict complementarity if one exists. On the other hand, if the primal feasibility is reduced too slowly, then the algorithm converges to a point of minimal complementarity; if the primal feasibility is reduced too quickly and the set of Lagrange multipliers is unbounded, then the norm of the Lagrange multiplier tends to infinity. Our theory has important implications for the design of IPMs. Specifically, we show that IPOPT, an algorithm that does not carefully control primal feasibility has practical issues with the dual multipliers values growing to unnecessarily large values. Conversely, the one-phase IPM of \citet*{hinder2018one}, an algorithm that controls primal feasibility as our theory suggests, has no such issue.

Communication Complexity of Dual Decomposition Methods for Distributed Resource Allocation Optimization

Dual decomposition methods are among the most prominent approaches for finding primal/dual saddle point solutions of resource allocation optimization problems. To deploy these methods in the emerging Internet of things networks, which will often have limited data rates, it is important to understand the communication overhead they require. Motivated by this, we introduce and explore two measures of communication complexity of dual decomposition methods to identify the most efficient communication among these algorithms. The first measure is $\epsilon$ -complexity, which quantifies the minimal number of bits needed to find an $\epsilon$ -accurate solution. The second measure is $b$ -complexity, which quantifies the best possible solution accuracy that can be achieved from communicating $b$ bits. We find the exact $\epsilon$ - and $b$ -complexity of a class of resource allocation problems where a single supplier allocates resources to multiple users. For both the primal and dual problems, the $\epsilon$ -complexity grows proportionally to $\log _2(1/\epsilon)$ and the $b$ -complexity proportionally to $1/2^b$ . We also introduce a variant of the $\epsilon$ - and $b$ -complexity measures where only algorithms that ensure primal feasibility of the iterates are allowed. Such algorithms are often desirable because overuse of the resources can overload the respective systems, e.g., by causing blackouts in power systems. We provide upper and lower bounds on the convergence rate of these primal feasible complexity measures. In particular, we show that the $b$ -complexity cannot converge at a faster rate than $\mathcal {O}(1/b)$ . Therefore, the results demonstrate a tradeoff between fast convergence and primal feasibility. We illustrate the result by numerical studies.

Optimal portfolio execution under time-varying liquidity constraints

ABSTRACTIn this article, we take an algorithmic approach to solve the problem of optimal execution under time-varying constraints on the depth of a limit order book (LOB). Our algorithms are within the resilience model proposed by Obizhaeva and Wang (2013) with a more realistic assumption on the order book depth; the amount of liquidity provided by an LOB market is finite at all times. For the simplest case where the order book depth stays at a fixed level for the entire trading horizon, we reduce the optimal execution problem into a one-dimensional root-finding problem which can be readily solved by standard numerical algorithms. When the depth of the order book is monotone in time, we apply the Karush-Kuhn-Tucker conditions to narrow down the set of candidate strategies. Then, we use a dichotomy-based search algorithm to pin down the optimal one. For the general case, we start from the optimal strategy subject to no liquidity constraints and iterate over execution strategy by sequentially adding more constraints to the problem in a specific fashion until primal feasibility is achieved. Numerical experiments indicate that our algorithms give comparable results to those of current existing convex optimization toolbox CVXOPT with significantly lower time complexity.

Strongly polynomial primal monotonic build-up simplex algorithm for maximal flow problems

The maximum flow problem (MFP) is a fundamental model in operations research. The network simplex algorithm is one of the most efficient solution methods for MFP in practice. The theoretical properties of established pivot algorithms for MFP are less understood. Variants of the primal simplex and dual simplex methods for MFP have been proven strongly polynomial, but no similar result exists for other pivot algorithms like the monotonic build-up or the criss-cross simplex algorithm.The monotonic build-up simplex algorithm (MBU SA) starts with a feasible solution, and fixes the dual feasibility one variable at a time, temporarily losing primal feasibility. In the case of maximum flow problems, pivots in one such iteration are all dual degenerate, bar the last one. Using a labelling technique to break these ties we show a variant that solves the maximum flow problem in 2|V||E|2 pivots.

Iteration complexity analysis of dual first-order methods for conic convex programming

In this paper we provide a detailed analysis of the iteration complexity of dual first-order methods for solving conic convex problems. When it is difficult to project on the primal feasible set described by conic and convex constraints, we use the Lagrangian relaxation to handle the conic constraints and then, we apply dual first-order algorithms for solving the corresponding dual problem. We give convergence analysis for dual first-order algorithms (dual gradient and fast gradient algorithms): we provide sublinear or linear estimates on the primal suboptimality and feasibility violation of the generated approximate primal solutions. Our analysis relies on the Lipschitz property of the gradient of the dual function or an error bound property of the dual. Furthermore, the iteration complexity analysis is based on two types of approximate primal solutions: the last primal iterate or an average primal sequence.

Dual Inequalities for Stabilized Column Generation Revisited

Column generation (CG) models have several advantages over compact formulations: they provide better linear program bounds, may eliminate symmetry, and can hide nonlinearities in their subproblems. However, users also encounter drawbacks in the form of slow convergence, also known as the tailing-off effect, and the oscillation of the dual variables. Among different alternatives for stabilizing the CG process, Ben Amor et al. [Ben Amor H, Desrosiers J, Valério de Carvalho JM (2006) Dual-optimal inequalities for stabilized column generation. Oper. Res. 54(3):454–463] suggest the use of dual-optimal inequalities (DOIs) in the context of cutting stock and bin packing problems. We generalize their results, provide new classes of (deep) DOIs, and show the applicability to other problems (vector packing, vertex coloring, bin packing with conflicts). We also suggest the dynamic addition of violated dual inequalities in a cutting-plane fashion and the use of dual inequalities that are not necessarily (deep) DOIs. In the latter case, a recovery procedure is needed to restore primal feasibility. Computational results proving the usefulness of the methods are presented.

Fast inexact decomposition algorithms for large-scale separable convex optimization

In this paper, we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The algorithm has low computational complexity since it consists in only one primal step and two dual steps at each iteration and allows one to solve the subproblem of each component inexactly and in parallel. Moreover, the algorithmic parameters are updated automatically without any tuning strategy as it happens in augmented Lagrangian approaches. We analyse the convergence of the algorithm and estimate its analytical worst-case complexity for both the primal–dual suboptimality and the primal feasibility violation, where is a given accuracy. Extensive numerical tests confirm that our method is numerically more efficient than the classical decomposition methods from the literature.

On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems

In this paper we propose a fully distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence. In particular, we prove that under the assumption of strong convexity and Lipschitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem. Using this error bound property we devise a fully distributed dual gradient scheme, i.e. a gradient scheme based on a weighted step size, for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation. Numerical simulations are also provided to confirm our theory.

Exterior point simplex-type algorithms for linear and network optimization problems

Two decades of research led to the development of a number of efficient algorithms that can be classified as exterior point simplex-type. This type of algorithms can cross over the infeasible region of the primal (dual) problem and find an optimal solution reducing the number of iterations needed. The main idea of exterior point simplex-type algorithms is to compute two paths/flows. Primal (dual) exterior point simplex-type algorithms compute one path/flow which is basic but not always primal (dual) feasible and the other is primal (dual) feasible but not always basic. The aim of this paper is to explain to the general OR audience, for the first time, the developments in exterior point simplex-type algorithms for linear and network optimization problems, over the recent years. We also present other approaches that, in a similar way, do not preserve primal or dual feasibility at each iteration such as the monotonic build-up Simplex algorithms and the criss-cross methods.

About the minimum mean cycle-canceling algorithm

This paper focuses on the resolution of the capacitated minimum cost flow problem on a network comprising n nodes and m arcs. We present a method that counts imperviousness to degeneracy among its strengths, namely the minimum mean cycle-canceling algorithm (MMCC). At each iteration, primal feasibility is maintained and the objective function strictly improves. The goal is to write a uniform and hopefully more accessible paper which centralizes the ideas presented in the seminal work of Goldberg and Tarjan (1989) as well as the additional paper of Radzik and Goldberg (1994) where the complexity analysis is refined. Important properties are proven using linear programming rather than constructive arguments.We also retrieve Cancel-and-Tighten from the former paper, where each so-called phase which can be seen as a group of iterations requires O(mlogn) time. MMCC turns out to be a strongly polynomial algorithm which runs in O(mn) phases, hence in O(m2nlogn) time. This new complexity result is obtained with a combined analysis of the results in both papers along with original contributions which allows us to enlist Cancel-and-Tighten as an acceleration strategy.

An adaptive constraint tightening approach to linear model predictive control based on approximation algorithms for optimization

SummaryIn this paper, we propose a model predictive control scheme for discrete‐time linear invariant systems based on inexact numerical optimization algorithms. We assume that the solution of the associated quadratic program produced by some numerical algorithm is possibly neither optimal nor feasible, but the algorithm is able to provide estimates on primal suboptimality and primal feasibility violation. By adaptively tightening the complicating constraints, we can ensure the primal feasibility of the approximate solutions generated by the algorithm. We derive a control strategy that has the following properties: the constraints on the states and inputs are satisfied, asymptotic stability of the closed‐loop system is guaranteed, and the number of iterations needed for a desired level of suboptimality can be determined. The proposed method is illustrated using a simulated longitudinal flight control problem. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Projected gradient algorithms for optimization over order simplices

A primal–dual algorithm is proposed for computing the distance from a point to an order simplex. An advantage of the algorithm is that, for any initial active set, it can adjust the active set to improve both primal and dual feasibility until the optimal active set is found. We verify that the algorithm takes only O(n) elementary arithmetic operations, where n is the problem dimension. Numerical results demonstrate the efficiency of the primal–dual algorithm compared with the primal feasible algorithm and the dual feasible algorithm. The primal–dual algorithm proves very useful in projected gradient algorithms applied to general order simplex constrained problems since a series of projection subproblems are requested there and the primal–dual algorithm makes warm starts possible.