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Register a new userProportional-Integral Projected Gradient Method for Infeasibility Detection in Conic Optimization
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Added by
Yue Yu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A constrained optimization problem is primal infeasible if its constraints cannot be satisfied, and dual infeasible if the constraints of its dual problem cannot be satisfied. We propose a novel iterative method, named proportional-integral projected gradient method (PIPG), for detecting primal and dual infeasiblity in convex optimization with quadratic objective function and conic constraints. The iterates of PIPG either asymptotically provide a proof of primal or dual infeasibility, or asymptotically satisfy a set of primal-dual optimality conditions. Unlike existing methods, PIPG does not compute matrix inverse, which makes it better suited for large-scale and real-time applications. We demonstrate the application of PIPG in quasiconvex and mixed-integer optimization using examples in constrained optimal control.
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Conic optimization is the minimization of a differentiable convex objective function subject to conic constraints. We propose a novel primal-dual first-order method for conic optimization, named proportional-integral projected gradient method (PIPG). PIPG ensures that both the primal-dual gap and the constraint violation converge to zero at the rate of (O(1/k)), where (k) is the number of iterations. If the objective function is strongly convex, PIPG improves the convergence rate of the primal-dual gap to (O(1/k^2)). Further, unlike any existing first-order methods, PIPG also improves the convergence rate of the constraint violation to (O(1/k^3)). We demonstrate the application of PIPG in constrained optimal control problems.
This paper studies the distributed optimization problem where the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of the local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective functions approximate subgradients. We first prove that the formulated optimization problem can only be solved with an approximate error depending upon the accuracy of the available subgradients. Then, we show the exact solvability of this optimization problem if the accumulated approximation error is not too large. After that, we also give a novel componentwise normalized variant to improve the transient behavior of the convergent sequence. The effectiveness of our algorithms is verified by a numerical example.
We study the problem of detecting infeasibility of large-scale linear programming problems using the primal-dual hybrid gradient method (PDHG) of Chambolle and Pock (2011). The literature on PDHG has mostly focused on settings where the problem at hand is assumed to be feasible. When the problem is not feasible, the iterates of the algorithm do not converge. In this scenario, we show that the iterates diverge at a controlled rate towards a well-defined ray. The direction of this ray is known as the infimal displacement vector $v$. The first contribution of our work is to prove that this vector recovers certificates of primal and dual infeasibility whenever they exist. Based on this fact, we propose a simple way to extract approximate infeasibility certificates from the iterates of PDHG. We study three different sequences that converge to the infimal displacement vector: the difference of iterates, the normalized iterates, and the normalized average. All of them are easy to compute, and thus the approach is suitable for large-scale problems. Our second contribution is to establish tight convergence rates for these sequences. We demonstrate that the normalized iterates and the normalized average achieve a convergence rate of $O(1/k)$, improving over the known rate of $O(1/sqrt{k})$. This rate is general and applies to any fixed-point iteration of a nonexpansive operator. Thus, it is a result of independent interest since it covers a broad family of algorithms, including, for example, ADMM, and can be applied settings beyond linear programming, such as quadratic and semidefinite programming. Further, in the case of linear programming we show that, under nondegeneracy assumptions, the iterates of PDHG identify the active set of an auxiliary feasible problem in finite time, which ensures that the difference of iterates exhibits eventual linear convergence to the infimal displacement vector.
In this paper, we propose a relaxation to the stochastic ruler method originally described by Yan and Mukai in 1992 for asymptotically determining the global optima of discrete simulation optimization problems. We show that our proposed variant of the stochastic ruler method provides accelerated convergence to the optimal solution by providing computational results for two example problems, each of which support the better performance of the variant of the stochastic ruler over the original. We then provide the theoretical grounding for the asymptotic convergence in probability of the variant to the global optimal solution under the same set of assumptions as those underlying the original stochastic ruler method.
Distributed optimization is concerned with using local computation and communication to realize a global aim of optimizing the sum of local objective functions. It has gained wide attention for a variety of applications in networked systems. This paper addresses a class of constrained distributed nonconvex optimization problems involving univariate objective functions, aiming to achieve global optimization without requiring local evaluations of gradients at every iteration. We propose a novel algorithm named CPCA, exploiting the notion of combining Chebyshev polynomial approximation, average consensus and polynomial optimization. The proposed algorithm is i) able to obtain $epsilon$ globally optimal solutions for any arbitrarily small given accuracy $epsilon$, ii) efficient in terms of both zeroth-order queries (i.e., evaluations of function values) and inter-agent communication, and iii) distributed terminable when the specified precision requirement is met. The key insight is to use polynomial approximations to substitute for general objective functions, and turn to solve an easier approximate version of the original problem. Due to the nice analytic properties owned by polynomials, this approximation not only facilitates efficient global optimization, but also allows the design of gradient-free iterations to reduce cumulative costs of queries and achieve geometric convergence when nonconvex problems are solved. We provide comprehensive analysis of the accuracy and complexities of the proposed algorithm.
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