In this paper we consider the unconstrained minimization problem of a smooth function in ${\mathbb{R}}^n$ in a setting where only function evaluations are possible. We design a novel randomized derivative-free algorithm --- the stochastic three points (STP) method --- and analyze its iteration complexity. At each iteration, STP generates a random search direction according to a certain fixed probability law. Our assumptions on this law are very mild: roughly speaking, all laws which do not concentrate all measure on any halfspace passing through the origin will work. For instance, we allow for the uniform distribution on the sphere and also distributions that concentrate all measure on a positive spanning set.
Given a current iterate $x$, STP compares the objective function at three points: $x$, $x+αs$ and $x-αs$, where $α>0$ is a stepsize parameter and $s$ is the random search direction. The best of these three points is the next iterate. We analyze the method STP under several stepsize selection schemes (fixed, decreasing, estimated through finite differences, etc). We study non-convex, convex and strongly convex cases.
In this paper, based on the limited memory techniques and subspace minimization conjugate gradient (SMCG) methods, a regularized limited memory subspace minimization conjugate gradient method is proposed, which contains two types of iterations. In SMCG iteration, we obtain the search direction by minimizing the approximate quadratic model or approximate regularization model. In RQN iteration, combined with regularization technique and BFGS method, a modified regularized quasi-Newton method is used in the subspace to improve the orthogonality. Moreover, some simple acceleration criteria and an …
Subspace minimization conjugate gradient (SMCG) methods have become a class of quite efficient iterative methods for unconstrained optimization and have attracted extensive attention recently. Usually, the search directions of SMCG methods are generated by minimizing approximate models with the approximation matrix $ B_k $ of the objective function at the current iterate over the subspace spanned by the current gradient $ g_k $ and the latest search direction. The $ g_k^TB_kg_k $ must be estimated properly in the calculation of the search directions, which is crucial to the theoretical propert…
We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer minimizers as well. In case this condition holds, we use sos programming to compute the radius of a p-norm ball which contains all integer minimizers. We prove that this radius is smaller than the radius known from the literature. Furthermore, we derive a new class of underestimators of the polynomial function. Using a Stellensatz from real algebraic geometry and a…
In this paper, two new subspace minimization conjugate gradient methods based on $p - $regularization models are proposed, where a special scaled norm in $p - $regularization model is analyzed. Different choices for special scaled norm lead to different solutions to the $p - $regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. $R - $linear convergence of the pr…
We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to achieve a lower rank solution. This procedure is computationally efficient as it does not require projecting on the cone of positive-semidefinite matrices. Its performance in terms of objective improvement and rank reduction is tested over multiple graphs of large-scale Gset graph collection and over binary optimization problems from the Biq Mac collection.
Mohammed VI Polytechnic University（穆罕默德六世理工大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
In this paper, we propose a new zero order optimization method called minibatch stochastic three points (MiSTP) method to solve an unconstrained minimization problem in a setting where only an approximation of the objective function evaluation is possible. It is based on the recently proposed stochastic three points (STP) method (Bergou et al., 2020). At each iteration, MiSTP generates a random search direction in a similar manner to STP, but chooses the next iterate based solely on the approximation of the objective function rather than its exact evaluations. We also analyze our method's comp…
In this paper we study general $l_p$ regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of first- and second-order stationary points, and hence also of local minimizers of the $l_p$ minimization problems. We extend some existing iterative reweighted $l_1$ (IRL1) and $l_2$ (IRL2) minimization methods to solve these problems and proposed new variants for them in which each subproblem has a closed form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous $ε$-approxi…
Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning or batch Bayesian optimization, the objective function is not exactly submodular, but close. In this case, no theoretical guarantees exist. Indeed, submodular minimization algorithms rely on intricate connections between submodularity and convexity. We show how these relations can be extended to obtain approximation guarantees for minimizing non-submodular functions, characterized by how close the function is t…
The aim of this paper is to develop an efficient algorithm for solving a class of unconstrained nondifferentiable convex optimization problems in finite dimensional spaces. To this end we formulate first its Fenchel dual problem and regularize it in two steps into a differentiable strongly convex one with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method with the aim of accelerating the resulting convergence scheme. The theoretical results are finally applied to an l1 regularization problem arising in image processing.
In this article we investigate the possibilities of accelerating the double smoothing technique when solving unconstrained nondifferentiable convex optimization problems. This approach relies on the regularization in two steps of the Fenchel dual problem associated to the problem to be solved into an optimization problem having a differentiable strongly convex objective function with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method. The aim of this paper is to show how do the properties of the functions in the objective of the primal …
We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton's method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity results to be proved for convergence to points that satisfy approximate first-order and second-order optimality conditions. The complexity results match the best known results in…
In this paper we propose an augmented smoothing function for nonlinear L1 -norm minimization problem and consider a global stability of a gradient-based neural network model to minimize the smoothing function. The numerical simulations show that our smoothing neural network finds successfully the global solution of the L1 -norm minimization problems considered in the simulation.
We obtain a new lower bound on the information-based complexity of first-order minimization of smooth and convex functions. We show that the bound matches the worst-case performance of the recently introduced Optimized Gradient Method, thereby establishing that the bound is tight and can be realized by an efficient algorithm. The proof is based on a novel construction technique of smooth and convex functions.
This paper presents smoothing schemes for obtaining approximate stationary points of unconstrained or linearly-constrained composite nonconvex-concave min-max (and hence nonsmooth) problems by applying well-known algorithms to composite smooth approximations of the original problems. More specifically, in the unconstrained (resp. constrained) case, approximate stationary points of the original problem are obtained by applying, to its composite smooth approximation, an accelerated inexact proximal point (resp. quadratic penalty) method presented in a previous paper by the authors. Iteration com…
We construct a family of functions suitable for establishing lower bounds on the oracle complexity of first-order minimization of smooth strongly-convex functions. Based on this construction, we derive new lower bounds on the complexity of strongly-convex minimization under various inaccuracy criteria. The new bounds match the known upper bounds up to a constant factor, and when the inaccuracy of a solution is measured by its distance to the solution set, the new lower bound exactly matches the upper bound obtained by the recent Information-Theoretic Exact Method by the same authors, thereby e…
In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for the non convex case, of the inexact proximal method for multiobjective convex minimization problems studied by Bonnel et al. (SIAM Journal on Optimization 15, 4, 953-970, 2005).
In this paper, we construct an algorithm for minimising piecewise smooth functions for which derivative information is not available. The algorithm constructs a pair of quadratic functions, one on each side of the point with smallest known function value, and selects the intersection of these quadratics as the next test point. This algorithm relies on the quadratic function underestimating the true function within a specific range, which is accomplished using a adjustment term that is modified as the algorithm progresses.
We introduce new optimized first-order methods for smooth unconstrained convex minimization. Drori and Teboulle recently described a numerical method for computing the $N$-iteration optimal step coefficients in a class of first-order algorithms that includes gradient methods, heavy-ball methods, and Nesterov's fast gradient methods. However, Drori and Teboulle's numerical method is computationally expensive for large $N$, and the corresponding numerically optimized first-order algorithm requires impractical memory and computation for large-scale optimization problems. In this paper, we propose…
In this paper, the convergence of alternating minimization is established for non-smooth convex optimization in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth and a non-smooth part is considered with the latter being block-separable, e.g., corresponding to convex constraints or regularization. For the smooth part, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. Linear convergence is established for the first two cases, generalizin…
Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [IFF 2001, IO 2009]. However, these algorithms are typically not practical. In 1976, Wolfe proposed an algorithm to find the minimum Euclidean norm point in a polytope, and in 1980, Fujishige showed how Wolfe's algorithm can be used for SFM. For general submodular functions, this Fujishige-Wolfe minimum norm algorithm seems to have the best empirical performance. Despite its goo…
We consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with an linear mapping. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many…
An Adagrad-inspired class of algorithms for smooth unconstrained optimization is presented in which the objective function is never evaluated and yet the gradient norms decrease at least as fast as $\calO(1/\sqrt{k+1})$ while second-order optimality measures converge to zero at least as fast as $\calO(1/(k+1)^{1/3})$. This latter rate of convergence is shown to be essentially sharp and is identical to that known for more standard algorithms (like trust-region or adaptive-regularization methods) using both function and derivatives' evaluations. A related "divergent stepsize" method is also desc…
We introduce and analyze an algorithm for the minimization of convex functions that are the sum of differentiable terms and proximable terms composed with linear operators. The method builds upon the recently developed smoothed gap technique. In addition to a precise convergence rate result, valid even in the presence of linear inclusion constraints, this new method allows an explicit treatment of the gradient of differentiable functions and can be enhanced with line-search. We also study the consequences of restarting the acceleration of the algorithm at a given frequency. These new features …
We propose a conditional gradient framework for a composite convex minimization template with broad applications. Our approach combines smoothing and homotopy techniques under the CGM framework, and provably achieves the optimal $\mathcal{O}(1/\sqrt{k})$ convergence rate. We demonstrate that the same rate holds if the linear subproblems are solved approximately with additive or multiplicative error. In contrast with the relevant work, we are able to characterize the convergence when the non-smooth term is an indicator function. Specific applications of our framework include the non-smooth mini…
This paper describes an extension of the BFGS and L-BFGS methods for the minimization of a nonlinear function subject to errors. This work is motivated by applications that contain computational noise, employ low-precision arithmetic, or are subject to statistical noise. The classical BFGS and L-BFGS methods can fail in such circumstances because the updating procedure can be corrupted and the line search can behave erratically. The proposed method addresses these difficulties and ensures that the BFGS update is stable by employing a lengthening procedure that spaces out the points at which gr…
We consider the exact continuous relaxation model of matrix rank minimization problem proposed by Yu and Zhang (Comput.Optim.Appl. 1-20, 2022). Motivated by the inertial techinique, we propose a general inertial smoothing proximal gradient algorithm(GIMSPG) for this kind of problems. It is shown that the singular values of any accumulation point have a common support set and the nonzero singular values have a unified lower bound. Besides, the zero singular values of the accumulation point can be achieved within finite iterations. Moreover, we prove that any accumulation point of the sequence g…
The composite $L_q~(0<q<1)$ minimization problem over a general polyhedron has received various applications in machine learning, wireless communications, image restoration, signal reconstruction, etc. This paper aims to provide a theoretical study on this problem. Firstly, we show that for any fixed $0<q<1$, finding the global minimizer of the problem, even its unconstrained counterpart, is strongly NP-hard. Secondly, we derive Karush-Kuhn-Tucker (KKT) optimality conditions for local minimizers of the problem. Thirdly, we propose a smoothing sequential quadratic programming framework for solv…
This paper generalizes the optimized gradient method (OGM) that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound with rate $O(1/N^{1.5})$ on the decrease of the gradient norm among fixed-step first-order methods. Th…
A regularization algorithm (AR1pGN) for unconstrained nonlinear minimization is considered, which uses a model consisting of a Taylor expansion of arbitrary degree and regularization term involving a possibly non-smooth norm. It is shown that the non-smoothness of the norm does not affect the $O(ε_1^{-(p+1)/p})$ upper bound on evaluation complexity for finding first-order $ε_1$-approximate minimizers using $p$ derivatives, and that this result does not hinge on the equivalence of norms in $\Re^n$. It is also shown that, if $p=2$, the bound of $O(ε_2^{-3})$ evaluations for finding second-order …
We formulate an affine invariant implementation of the accelerated first-order algorithm in Nesterov (1983). Its complexity bound is proportional to an affine invariant regularity constant defined with respect to the Minkowski gauge of the feasible set. We extend these results to more general problems, optimizing Hölder smooth functions using $p$-uniformly convex prox terms, and derive an algorithm whose complexity better fits the geometry of the feasible set and adapts to both the best Hölder smoothness parameter and the best gradient Lipschitz constant. Finally, we detail matching complexity…
In this work, we consider a constrained convex problem with linear inequalities and provide an inexact penalty re-formulation of the problem. The novelty is in the choice of the penalty functions, which are smooth and can induce a non-zero penalty over some points in feasible region of the original constrained problem. The resulting unconstrained penalized problem is parametrized by two penalty parameters which control the slope and the curvature of the penalty function. With a suitable selection of these penalty parameters, we show that the solutions of the resulting penalized unconstrained p…
We introduce a novel approach for analyzing the performance of first-order black-box optimization methods. We focus on smooth unconstrained convex minimization over the Euclidean space $R^d$. Our approach relies on the observation that by definition, the worst case behavior of a black-box optimization method is by itself an optimization problem, which we call the Performance Estimation Problem (PEP). We formulate and analyze the PEP for two classes of first-order algorithms. We first apply this approach on the classical gradient method and derive a new and tight analytical bound on its perform…
We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions on the problem template. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best known convergence rate results, in particular when the template consists of only non-smooth functions. We also outline a restart strat…
In this paper, an equivalent smooth minimization for the L1 regularized least square problem is proposed. The proposed problem is a convex box-constrained smooth minimization which allows applying fast optimization methods to find its solution. Further, it is investigated that the property "the dual of dual is primal" holds for the L1 regularized least square problem. A solver for the smooth problem is proposed, and its affinity to the proximal gradient is shown. Finally, the experiments on L1 and total variation regularized problems are performed, and the corresponding results are reported.
