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A Level-Set Method for Convex Optimization with a 2 Feasible Solution Path

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A Level-Set Method for Convex Optimization with a 2 Feasible Solution Path 1 A LEVEL-SET METHOD FOR CONVEX OPTIMIZATION WITH A 2 FEASIBLE SOLUTION PATH 3 QIHANG LIN∗, SELVAPRABU NADARAJAHy , AND NEGAR SOHEILIz 4 Abstract. Large-scale constrained convex optimization problems arise in several application 5 domains. First-order methods are good candidates to tackle such problems due to their low iteration 6 complexity and memory requirement. The level-set framework extends the applicability of first-order 7 methods to tackle problems with complicated convex objectives and constraint sets. Current methods 8 based on this framework either rely on the solution of challenging subproblems or do not guarantee a 9 feasible solution, especially if the procedure is terminated before convergence. We develop a level-set 10 method that finds an -relative optimal and feasible solution to a constrained convex optimization 11 problem with a fairly general objective function and set of constraints, maintains a feasible solution 12 at each iteration, and only relies on calls to first-order oracles. We establish the iteration complexity 13 of our approach, also accounting for the smoothness and strong convexity of the objective function 14 and constraints when these properties hold. The dependence of our complexity on is similar to 15 the analogous dependence in the unconstrained setting, which is not known to be true for level-set 16 methods in the literature. Nevertheless, ensuring feasibility is not free. The iteration complexity of 17 our method depends on a condition number, while existing level-set methods that do not guarantee 18 feasibility can avoid such dependence. We numerically validate the usefulness of ensuring a feasible 19 solution path by comparing our approach with an existing level set method on a Neyman-Pearson classification problem.20 21 Key words. constrained convex optimization, level-set technique, first-order methods, com- plexity analysis22 AMS subject classifications. 90C25, 90C30, 90C5223 24 1. Introduction. Large scale constrained convex optimization problems arise 25 in several business, science, and engineering applications. A commonly encountered 26 form for such problems is 27 f ∗ := min f(x)(1) x2X 2928 (2) s.t. gi(x) ≤ 0; i = 1; : : : ; m; n 30 where X ⊆ R is a non-empty closed and convex set, and f and gi, i = 1; : : : ; m, are 31 closed convex real-valued functions defined on X . Given > 0, a point x is -optimal ∗ 32 if it satisfies f(x ) − f ≤ and -feasible if it satisfies maxi=1;:::;m gi(x ) ≤ . In 33 addition, given 0 < ≤ 1, and a feasible and suboptimal solution e 2 X , a point x is 34 -relative optimal with respect to e if it satisfies (f(x) − f ∗)=(f(e) − f ∗) ≤ . 35 Level-set techniques [2,3, 15, 24, 39, 40] provide a flexible framework for solving 36 the optimization problem (1)-(2). Specifically, this problem can be reformulated as a ∗ ∗ 37 convex root finding problem L (t) = 0, where L (t) = minx2X maxff(x)−t; gi(x); i = 38 1; : : : ; mg (see Lemar´echal et al. [24]). Then a bundle method [21, 23, 41] or an accel- 39 erated gradient method [29, Section 2.3.5] can be applied to handle the minimization 40 subproblem to estimate L∗(t) and facilitate the root finding process. In general, these 41 methods solve the subproblem using a nontrivial quadratic program that needs to be 42 solved at each iteration. 43 An important variant of (1)-(2) arising in statistics, image processing, and signal 44 processing occurs when f(x) takes a simple form, such as a gauge function, and there ∗Tippie College of Business, University of Iowa, USA, [email protected] yCollege of Business Administration, University of Illinois at Chicago, USA, [email protected] zCollege of Business Administration, University of Illinois at Chicago, USA, [email protected] 1 This manuscript is for review purposes only. 2 QIHANG LIN, SELVAPRABU NADARAJAH, AND NEGAR SOHEILI 45 is a single constraint with the structure ρ(Ax − b) ≤ λ, where ρ is a convex function, 46 A is a matrix, b is a vector, and λ is a scalar. In this setting, Van Den Berg and 47 Friedlander [39, 40] and Aravkin et al. [2] use a level-set approach to swap the roles 48 of the objective function and constraints in (1)-(2) to obtain the convex root finding 49 problem v(t) = 0 with v(t) := minx2X ;f(x)≤t ρ(Ax − b) − λ. Harchaoui et al. [15] 50 proposed a similar approach. The root finding problem arising in this approach is 51 solved using inexact newton and secant methods in [2, 39, 40] and [2], respectively. 52 Since f(x) is assumed to be simple, projection onto the feasible set fx 2 Rnjx 2 53 X ; f(x) ≤ tg is possible and the subproblem in the definition of v(t) can be solved 54 by various first-order methods. Although the assumption on the objective function 55 is important, the constraint structure is not limiting because (2) can be obtained by 56 choosing A to be an identity matrix, b = 0, λ = 0, and ρ(x) = maxi=1;:::;m gi(x). 57 The level-set techniques discussed above find an -feasible and -optimal solution 58 to (1)-(2), that is, the terminal solution may not be feasible. This issue can be 59 overcome when the infeasible terminal solution is also super-optimal and a strictly 60 feasible solution to the problem exists. In this case, a radial-projection scheme [2, 35] 61 can be applied to obtain a feasible and -relative optimal solution. Super-optimality of 62 the terminal solution holds if the constraint f(x) ≤ t is satisfied at termination. This 63 condition can be enforced when projections on to the feasible set with this constraint 64 are computationally cheap, which typically requires f(x) to be simple (e.g. see [2]). 65 Otherwise, such a guarantee does not exist. 66 Building on this rich literature, we develop a feasible level-set method that (i) 67 employs a first-order oracle to solve subproblems and compute a feasible and -relative 68 optimal solution to (1)-(2) with a general objective function and set of constraints 69 while also maintaining feasible intermediate solutions at each iteration (i.e., a feasible 70 solution path), and (ii) it avoids potentially costly projections. Our method tackles 71 a root finding problem involving the univariate function L∗(t), that is L∗(t) = 0, 72 which differs from the univariate function v(t) considered by the level set methods 73 in [2, 39, 40]. The level set methods in [18] and [6] also solve L∗(t) = 0. However, 74 exact computation of L∗(t) is assumed in the former case while the latter paper 75 solves a potentially expensive quadratic problem when m is large to obtain inexact 76 evaluations of this function. Level set methods in the literature increase the level 77 parameter t when moving towards the root whereas our procedure approaches the 78 root by gradually decreasing t. This distinction is important to ensure feasibility. 79 Our focus on maintaining feasibility is especially desirable in situations where 80 level-set methods need to be terminated before convergence, for instance due to a 81 time constraint. In this case, our level-set approach would return a feasible, albeit 82 possibly suboptimal, solution that can be implemented whereas the solution from 83 existing level-set approaches may not be implementable due to potentially large con- 84 straint violations. Convex optimization problems where constraints model operating 85 conditions that need to be satisfied by solutions for implementation arise in several 86 applications, including network and nonlinear resource allocation, signal processing, 87 circuit design, and classification problems encountered in medical diagnosis and other 88 areas [10, 11, 33, 36]. We numerically validate the usefulness of ensuring feasibility on 89 a Neyman-Pearson classification problem, where we find that early termination of an 90 existing level set method can lead to highly infeasible solutions, while, as expected, 91 our feasible level set method provides a path of feasible solutions. 92 The radial sub-gradient method proposed by Renegar [35] and extended by Grim- 93 mer [14] also emphasizes feasibility. This method avoids projection and finds a feasible 94 and -relative optimal solution by performing a line search at each iteration. This line This manuscript is for review purposes only. 3 95 search, while easy to execute for some linear or conic programs, can be challenging for 96 general convex programs (1)-(2). Moreover, it is unknown whether the iteration com- 97 plexity of this approach can be improved by utilizing the potential smoothness and 98 strong convexity of the objective function and constraints. In contrast, the feasible 99 level-set approach in this paper avoids the need for line search, in addition to projec- 100 tion, and its iteration complexity leverages the strong convexity and smoothness of 101 the objective and constraints when these properties are true. 102 Our work is indeed related to first-order methods, which are popular due to their 103 low per-iteration cost and memory requirement, for example compared to interior 104 point methods, for solving large scale convex optimization problems [16]. Most ex- 105 isting first-order methods find a feasible and -optimal solution to a version of the 106 optimization problem (1)-(2) with a simple feasible region (e.g. a box, simplex or 107 ball) which makes projection on to the feasible set at each iteration inexpensive. En- 108 suring feasibility using projection becomes difficult for general constraints (2). Some 109 sub-gradient methods, including the method by Nesterov [29, Section 3.2.4] and its 110 recent variants by Lan and Zhou [22] and Bayandina [5], can be applied to (1)-(2) 111 without the need for projection mappings.
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