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We characterize the optimal solution to the linear programming relaxation of the standard formulation for the minimum satisfiability problem. We give a O ( n m 2 ) combinatorial algorithm to solve the fractional version of the minimum satisfiability problem optimally where n ( m ) is the number of variables (clauses). As a by-product, we obtain a 2 ( 1 − 1 ∕ 2 k ) approximation algorithm for the minimum satisfiability problem where k is the maximum number of literals in any clause.

In this note, we provide an overarching analysis of primal-dual dynamics associated to linear equality-constrained optimization problems using contraction analysis. For the well-known standard version of the problem: we establish convergence under convexity and the contracting rate under strong convexity. Then, for a canonical distributed optimization problem, we use partial contractivity to establish global exponential convergence of its primal-dual dynamics. As an application, we propose a new distributed solver for the least-squares problem with the same convergence guarantees. Finally, for time-varying versions of both centralized and distributed primal-dual dynamics, we exploit their contractive nature to establish bounds on their tracking error. To support our analyses, we introduce novel results on contraction theory.

In the k-means problem with penalties, we are given a data set $${\cal D} \subseteq \mathbb{R}^\ell $$ of n points where each point $$j \in {\cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set $${\rm{C}}S \subseteq {{\cal R}^\ell }$$ with |CS| ≤ k and a penalized subset $${{\cal D}_p} \subseteq {\cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely $$\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).

A new primal-dual interior-point algorithm applicable to nonsymmetric conic optimization is proposed. It is a generalization of the famous algorithm suggested by Nesterov and Todd for the symmetric conic case, and uses primal-dual scalings for nonsymmetric cones proposed by Tunçel. We specialize Tunçel’s primal-dual scalings for the important case of 3 dimensional exponential-cones, resulting in a practical algorithm with good numerical performance, on level with standard symmetric cone (e.g., quadratic cone) algorithms. A significant contribution of the paper is a novel higher-order search direction, similar in spirit to a Mehrotra corrector for symmetric cone algorithms. To a large extent, the efficiency of our proposed algorithm can be attributed to this new corrector.

In this paper we combine the stochastic variance reduced gradient (SVRG) method [17] with the primal dual fixed point method (PDFP) proposed in [7] to solve a sum of two convex functions and one of which is linearly composite. This type of problems are typically arisen in sparse signal and image reconstruction. The proposed SVRG-PDFP can be seen as a generalization of Prox-SVRG [37] originally designed for the minimization of a sum of two convex functions. Based on some standard assumptions, we propose two variants, one is for strongly convex objective function and the other is for general convex cases. Convergence analysis shows that the convergence rate of SVRG-PDFP is O(1/k) (here k is the iteration number) for general convex objective function and linear for k strongly convex case. Numerical examples on machine learning and CT image reconstruction are provided to show the effectiveness of the algorithms.

Problem definition: We study a fundamental online resource allocation problem in service operations in which a heterogeneous stream of arrivals that varies in service times and rewards makes service requests from a finite number of servers/providers. This is an online adversarial setting in which nothing more is known about the arrival process of customers. Each server has a finite regular capacity but can be expanded at the expense of overtime cost. Upon arrival of each customer, the system chooses both a server and a time for service over a scheduling horizon subject to capacity constraints. The system seeks easy-to-implement online policies that admit a competitive ratio (CR), guaranteeing the worst-case relative performance. Academic/practical relevance: On the academic side, we propose online algorithms with theoretical CRs for the problem described above. On the practical side, we investigate the real-world applicability of our methods and models on appointment-scheduling data from a partner health system. Methodology: We develop new online primal-dual approaches for making not only a server-date allocation decision for each arriving customer, but also an overtime decision for each server on each day within a horizon. We also derive a competitive analysis to prove a theoretical performance guarantee. Results: Our online policies are (i) robust to future information, (ii) easy-to-implement and extremely efficient to compute, and (iii) admitting a theoretical CR. Comparing our online policy with the optimal offline policy, we obtain a CR that guarantees the worst-case performance of our online policy. Managerial implications: We evaluate the performance of our online algorithms by using real appointment scheduling data from a partner health system. Our results show that the proposed online policies perform much better than their theoretical CR, and outperform the pervasive First-Come-First-Served (FCFS) and nested threshold policies (NTPO) by a large margin.

In this paper, we propose a second-order continuous primal-dual dynamical system with time-dependent positive damping terms for a separable convex optimization problem with linear equality constraints. By the Lyapunov function approach, we investigate asymptotic properties of the proposed dynamical system as the time $t\to+\infty$. The convergence rates are derived for different choices of the damping coefficients. We also show that the obtained results are robust under external perturbations.

We study assemble-to-order (ATO) problems from the literature. ATO problems with general structure and integrality constraints are well known to be difficult to solve, and we provide new insight into these issues by establishing worst-case approximation guarantees through primal-dual analyses and linear programming (LP) rounding. First, we relax the one-period ATO problem using a natural newsvendor decomposition and use the dual solution for the relaxation to derive a lower bound on optimal cost, providing a tight approximation guarantee that grows with the maximum product size in the system. Then, we present an LP rounding algorithm that achieves both asymptotic optimality as demand grows large, and a 1.8 approximation factor for any problem instance. In addition to theoretical guarantees, we perform comprehensive numerical simulations and find that our rounding algorithm outperforms existing techniques and is close to optimal. Finally, we demonstrate that our one-period LP rounding results can be used to develop an asymptotically optimal integral policy for dynamic ATO problems with backlogging and identical component lead-times. This paper was accepted by Yinyu Ye, optimization.

We consider a class of elliptic partial differential equations (PDE) that can be understood as the Euler–Lagrange equations of an associated convex optimization problem. Discretizing this optimization problem, we present a strategy for a numerical solution that is based on the popular primal-dual hybrid gradients (PDHG) approach: we reformulate the optimization as a saddle-point problem with a dual variable addressing the quadratic term, introduce the PDHG optimization steps, and analytically eliminate the dual variable. The resulting scheme resembles explicit gradient descent; however, the eliminated dual variable shows up as a boosting term that substantially accelerates the scheme. We introduce the proposed strategy for a simple Laplace problem and then illustrate the technique on a variety of more complicated and relevant PDE, both on Cartesian domains and graphs. The proposed numerical method is easily implementable, computationally efficient, and applicable to relevant computing tasks across science and engineering.

In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a K...

The$${\mathcal {S}}$$S-cone provides a common framework for cones of polynomials or exponential sums which establish non-negativity upon the arithmetic-geometric inequality, in particular for sums of non-negative circuit polynomials (SONC) or sums of arithmetic-geometric exponentials (SAGE). In this paper, we study the$${\mathcal {S}}$$S-cone and its dual from the viewpoint of second-order representability. Extending results of Averkov and of Wang and Magron on the primal SONC cone, we provide explicit generalized second-order descriptions for rational$${\mathcal {S}}$$S-cones and their duals.

As indicated in the recent studies about primal-dual interior-point methods (IPMs) based on kernel functions, a kernel function not only serves to determine the search direction and measure the distance of the current iteration point to the $$\mu $$ -center, but also affects the iteration complexity and the practical computational efficiency of the algorithm. In this paper, we propose a new IPM for semidefinite optimization (SDO) based on a parameterized kernel function which is a generalization of the one presented by Bai et al. (Optim Methods Softw 17(6):985–1008, 2002). By using the good properties of the parameterized kernel function, we deduce that the iteration bound for large-update method is $$O(\sqrt{n}\log {n}\log {\frac{n}{\epsilon }})$$ for $$q=O(n)$$ , which is the best known complexity results for such methods. In our knowledge, this result is the first instance of primal-dual interior point method for SDO which involving the kernel function. Some numerical results have been provided.

UKM Jovelyn is a small medium enterprise (SME) that engaged in the field of bakery production. Until now, they produce cakes and cookies. They sell products throughout the Batam area which includes (Tiban market, aviari market, mitra market and puja bahari market), shops, school canteens and so on. The SME targets several things such as profit per month, number of hours worked, number of overtime hours, machine utilities, labor utilities and so forth. Until now, the SME has not been able to reach the targets because the UKM has not determined the production amount based on the appropriate calculation. The number of products made is determined only on an intuitive basis. Determination of the amount of production cannot be determined by the usual simplex method because the objective function in the usual simplex method contains a target. Problem solving is done by using goal programming. The results of the calculation analysis show that to achieve the optimal value, the SME must produce coconut root cookies, butter cookies, nastar cookies, snow princess cookies and peanut cookies respectively 11.89 kg, 22.43 kg, 24.96 kg , 1.97kg and 0 kg. Sensitivity analysis is carried out after there are changes in conditions that result in changes in the linear program matrix, so the number of products that must be made also changes which include coconut root cookies as much as 12.8 kg, as much as 16.91 kg of butter cookies, nastar cookies as much as 25.7 and the last is 3.39 kg of nut cookies which in the initial calculation, nut cookies were not produced. The sensitivity analysis is not done by recalculation but rather by modifying the matrix and completing iteration with primal dual.

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming ...

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, then the algorithm must irrevocably either match it to an unmatched neighbor or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, and so on. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step toward solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, then we show a tight competitive ratio ≈0.5671 of Ranking. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 < 1- 1/e-competitive in our model, even for bipartite graphs.

As distributed energy resources increase, distributed optimization for power system operation and demand response (DR) have been intensively investigated in recent years. This paper presents a dist...

In this paper, we deal with a polynomial primal-dual interior-point algorithm for solving convex quadratic programming based on a new parametric kernel function with an exponential barrier term. Th...

We present a primal–dual modified log-barrier algorithm to solve inequality constrained nonlinear optimization problems. Basically, the algorithm is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing a modified log-barrier function to handle inequality constraints. The algorithm uses an outer/inner iteration scheme and the globalization is performed in the primal–dual space by means of a new primal–dual merit function. The robustness and efficiency of the algorithm is improved using quadratic extrapolation. The numerical performance of the new method is illustrated by comparing it with a primal–dual classical log-barrier method and two well-established interior-point solvers on two sets of problems from COPS and Hock–Schittkowski collections, including a set of problems that exhibits degeneracy.

In this paper, a new variant of accelerated gradient descent is proposed. The proposed method does not require any information about the objective function, uses exact line search for the practical accelerations of convergence, converges according to the well-known lower bounds for both convex and non-convex objective functions, possesses primal–dual properties and can be applied in the non-euclidian set-up. As far as we know this is the first such method possessing all of the above properties at the same time. We also present a universal version of the method which is applicable to non-smooth problems. We demonstrate how in practice one can efficiently use the combination of line-search and primal-duality by considering a convex optimization problem with a simple structure (for example, linearly constrained). © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group.

The work deal on a method for optimisation of 330KV power system load flow which excels other existing methods.This method is called, PRIMAL-DUAL INTERIOR-POINT TECHNIQUE for solving optimal load flow problem. As problems of load-shedding, power outages and system losses have been cause for worries, especially among the developing nations such as Nigeria, hence need for a load flow solution technique, which, this work addresses. Optimisation is achieving maximum of required and minimum of un-required and it is obtained mathematically by differentiating the objective function with respect to the control variable(s) and equating the resulting expression(s) to zero. In 330KV Power System, optimization is maximisation of real power injection, voltage magnitude and cost effectiveness, while minimization of reactive power injection, power loss, critical clearing time of fault conditions and time of load flow simulation.. This work developed a mathematical model that solves load flow problems by engaging non-negative PRIMAL variables, “S” and “z” into the inequality constraint of the load flow problems in other to transform it to equality constraint(s). Another non-negative DUAL variables “” and “v” are incorporated together with Lagrangian multiplier “λ” to solve optimisation. While solving optimisation Barrier Parameter “” which ensures feasible point(s) exist(s) within the feasible region (INTERIOR POINT). Damping factor or step length parameter “α”, in conjunction with Safety factor “” (which improves convergence and keeps the non-negative variables strictly positive) are employed to achieve result. The key-words which are capitalized joined to give this work its name, the PRIMAL-DUAL INTERIOR-POINT. The initial feasible point(s) is/are tested for convergence and where it/they fail(s), iteration starts. Variables are updated by using the computed step size and the step length parameter “α”, which thereafter, undergo another convergence test. This technique usually converges after first iteration. Primarily, this technique excels the existing methods as; it solves load flow problems with equality and inequality constraints simultaneously, it often converges after first iteration as against six or more iterations of the existing methods, its solution provides higher power generations from available capacity and minimum system loss. Example, Geregu Power Station on Bus 12 generates 0.1786p.u power from the available 0.2000p.u through the PD-IP tech. as against 0.1200p.u of the existing methods. Also it supplies 0.1750p.u with loss of 0.0036p.u as against 0.0236p.u with loss of 0.0964p.u of the existing methods. This results in 90% generation as against 60% of existing methods. Generation loss is 1.8% as against 80.3% of existing methods and availability loss of 12.5% as against 88.2% of existing. Therefore this method ensures very high system stability.

Online interval scheduling problems consider scheduling a sequence of jobs on machines to maximize the total reward. Various approaches and algorithms have been proposed for different problem formulations. This paper provides a primal-dual approach to analyze algorithms for online interval scheduling problems. This primal-dual technique can be used for both stochastic and adversarial job sequences, and hence, is universally and generally applicable. We use strong duality and complementary slackness conditions to derive exact algorithms for scheduling stochastic equal-length job sequences on a single machine. We use weak duality to obtain upper bounds for the optimal reward for scheduling stochastic equal-length job sequences on multiple machines and C-benevolent job sequences on a single machine.

In this paper, we propose the first hyperbolic-logarithmic kernel function for Semidefinite programming problems. By simple analysis tools, several properties of this kernel function are used to compute the total number of iterations. We show that the worst-case iteration complexity of our algorithm for large-update methods improves the obtained iteration bounds based on hyperbolic [24] as well as classic kernel functions. For small-update methods, we derive the best known iteration bound.

Electrical impedance tomography (EIT), geophysics and undersea target reconstruction are typical non-linear ill-posed inverse problems, and in many cases, the anomalous bodies feature with a clear boundary. Thus, it is suitable to obtain sharp boundaries and blocky features with the Total Variation (TV) functional regularization. However, the TV function regularization leads to a non-differentiable objective function at zero in the inverse formulation and reduces the algorithm robustness. In this paper, we propose an improved primal-dual interior-point method (PD-IPM) based on the Lawson norm to get sharp spatial profiles of the anomalous bodies. Furthermore, the impact of the smooth parameter is investigated to get the inverse results. Numerical experiment using simulated data is setup to support our claim.

In nonlinearly constrained optimization, penalty methods provide an effective strategy for handling equality constraints, while barrier methods provide an effective approach for the treatment of in...

This paper presents a primal-dual weak Galerkin (PD-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. The method is based on a variational form where all the derivatives are applied to the test functions so that no regularity is necessary for the exact solution of the model equation. The numerical scheme is designed by using locally constructed weak second order partial derivatives and the weak gradient commonly used in the weak Galerkin context. Optimal order of convergence is derived for the resulting numerical solutions. Numerical results are reported to demonstrate the performance of the numerical scheme., Comment: 26 pages, 10 tables, 3 figures

An antiwindup control framework is developed for primal-dual dynamics in convex optimization. The proposed architecture provides for straightforward implementation of the ensuing primal-dual dynamics and for establishing global asymptotic stability using the notion of shifted-passivity with Lyapunov function commonly encountered in antiwindup literature.