Mathematical Programming最新文献

We propose a novel trust region method for solving a class of nonsmooth, nonconvex composite-type optimization problems. The approach embeds inexact semismooth Newton steps for finding zeros of a normal map-based stationarity measure for the problem in a trust region framework. Based on a new merit function and acceptance mechanism, global convergence and transition to fast local q-superlinear convergence are established under standard conditions. In addition, we verify that the proposed trust region globalization is compatible with the Kurdyka–Łojasiewicz inequality yielding finer convergence results. Experiments on sparse logistic regression, image compression, and a constrained log-determinant problem illustrate the efficiency of the proposed algorithm.

We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of 3 for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any (b > 1) a tight analysis for the natural b-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of ((1+3sqrt{3})approx 6.197) for the deterministic and of (approx 3.032) for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is 2-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than 10 for adaptions of the b-scaling strategy to online release dates and unweighted jobs on identical parallel machines.

We study the convergence behaviors of primal–dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

Utility preference robust optimization (PRO) has recently been proposed to deal with optimal decision-making problems where the decision maker’s (DM’s) preference over gains and losses is ambiguous. In this paper, we take a step further to investigate the case that the DM’s preference is random. We propose to use a random utility function to describe the DM’s preference and develop distributional utility preference robust optimization (DUPRO) models when the distribution of the random utility function is ambiguous. We concentrate on data-driven problems where samples of the random parameters are obtainable but the sample size may be relatively small. In the case when the random utility functions are of piecewise linear structure, we propose a bootstrap method to construct the ambiguity set and demonstrate how the resulting DUPRO can be solved by a mixed-integer linear program. The piecewise linear structure is versatile in its ability to incorporate classical non-parametric utility assessment methods into the sample generation of a random utility function. Next, we expand the proposed DUPRO models and computational schemes to address general cases where the random utility functions are not necessarily piecewise linear. We show how the DUPRO models with piecewise linear random utility functions can serve as approximations for the DUPRO models with general random utility functions and allow us to quantify the approximation errors. Finally, we carry out some performance studies of the proposed bootstrap-based DUPRO model and report the preliminary numerical test results. This paper is the first attempt to use distributionally robust optimization methods for PRO problems.

A set of quadratic forms is simultaneously diagonalizable via congruence (SDC) if there exists a basis under which each of the quadratic forms is diagonal. This property appears naturally when analyzing quadratically constrained quadratic programs (QCQPs) and has important implications in globally solving such problems using branch-and-bound methods. This paper extends the reach of the SDC property by studying two new weaker notions of simultaneous diagonalizability. Specifically, we say that a set of quadratic forms is almost SDC (ASDC) if it is the limit of SDC sets and d-restricted SDC (d-RSDC) if it is the restriction of an SDC set in up to d-many additional dimensions. In the context of QCQPs, these properties correspond to problems that may be diagonalized after arbitrarily small perturbations or after the introduction of d additional variables. Our main contributions are complete characterizations of the ASDC pairs and nonsingular triples of symmetric matrices, as well as a sufficient condition for the 1-RSDC property for pairs of symmetric matrices. Surprisingly, we show that every singular symmetric pair is ASDC and that almost every symmetric pair is 1-RSDC. We accompany our theoretical results with preliminary numerical experiments applying these constructions to solve QCQPs within branch-and-bound schemes.

The reformulation–linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, we present a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs, which is based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Strategies for filtering product relations are discussed and tested. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. This algorithm is applicable to RLT cuts generated for all types of bilinear terms, including but not limited to the detected implicit products. A detailed computational study based on independent implementations in two solvers evaluates the performance impact of the proposed methods.

This paper considers approximation algorithms for generalized k-median problems. These problems can be informally described as k-median with a constant number of extra constraints, and includes k-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized k-median that outputs a 6.387-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for k-median with outliers as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018). The main technical innovation is allowing richer constraint sets in the iterative rounding and using the structure of the resulting extreme points. Using our pseudo-approximation algorithm, we give improved approximation algorithms for k-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios (6.994 + epsilon ) and (6.387 + epsilon ) for k-median with outliers and knapsack median, respectively. These improve on the best-known approximation ratio (7.081 + epsilon ) for both problems as reported (Krishnaswamy et al. in: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018).

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIAM J. Discrete Math. 29(1), 113–121 (2015)) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system ({xin mathbb {R}^n:, Ax=b,, mathbb {0}le xle u}) for (Ain mathbb {R}^{mtimes n}) is bounded by (O(mmin {m, n - m}log (m+kappa _A)+nlog n)), where (kappa _A) is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of A have polynomially bounded encoding length in n. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in (O(mn^2log (n+kappa _A))) augmentation steps.

Using the recently proposed maximal quadratic-free sets and the well-known monoidal strengthening procedure, we show how to improve intersection cuts for quadratically-constrained optimization problems by exploiting integrality requirements. We provide an explicit construction that allows an efficient implementation of the strengthened cuts along with computational results showing their improvements over the standard intersection cuts. We also show that, in our setting, there is unique lifting which implies that our strengthening procedure is generating the best possible cut coefficients for the integer variables.

Recently, convex nested stochastic composite optimization (NSCO) has received considerable interest for its applications in reinforcement learning and risk-averse optimization. However, existing NSCO algorithms have worse stochastic oracle complexities, by orders of magnitude, than those for simpler stochastic optimization problems without nested structures. Additionally, these algorithms require all outer-layer functions to be smooth, a condition violated by some important applications. This raises a question regarding whether the nested composition make stochastic optimization more difficult in terms of oracle complexity. In this paper, we answer the question by developing order-optimal algorithms for convex NSCO problems constructed from an arbitrary composition of smooth, structured non-smooth, and general non-smooth layer functions. When all outer-layer functions are smooth, we propose a stochastic sequential dual (SSD) method to achieve an oracle complexity of (mathcal {O}(1/epsilon ^2)) (resp., (mathcal {O}(1/epsilon ))) when the problem is convex (resp., strongly convex). If any outer-layer function is non-smooth, we propose a non-smooth stochastic sequential dual (nSSD) method to achieve an (mathcal {O}(1/epsilon ^2)) oracle complexity. We provide a lower complexity bound to show the latter (mathcal {O}(1/epsilon ^2)) complexity to be unimprovable, even under a strongly convex setting. All these complexity results seem to be new in the literature, and they indicate that convex NSCO problems have the same order of oracle complexity as problems without nested composition, except in the strongly convex and outer non-smooth cases.