An FPTAS for Minimizing a Class of Low-Rank Quasi-Concave Functions over a Convex Set ∗

We consider the problem of minimizing a class of quasi-concave functions over a convex set. Quasi-concave functions are a generalization of concave functions and thus, NP-hard to minimize over a convex set in general. We present a simple fully polynomial time approximation scheme (FPTAS) for minimizing a fairly general class of low-rank quasi-concave functions. Our algorithm is based on solving a polynomial number of linear minimization problems over a convex set and computes a near-optimal solution that is an extreme point of the convex set. Therefore, it applies directly to combinatorial 0 - 1 problems for which the convex hull of feasible solutions is known, such as shortest paths, spanning trees and matchings in undirected graphs.


Introduction
In this paper, we consider the problem of minimizing a class of low-rank quasi-concave functions over a convex set. Quasi-concave functions are a generalization of concave functions that arise in many important applications such as modeling economies of scale in inventory management and supply chain management problems through concave or quasi-concave procurement costs. Moreover, several important optimization problems can be formulated as concave minimization problems including 0-1 integer programming [25], quadratic assignment [17] and [3], bilinear programming [13] and linear complementarity problem [18]. Therefore, concave or quasi-concave minimization forms an important class of non-convex optimization problems that are computationally intractable in general.
In this paper, we consider the problem of minimizing a low-rank quasi-concave function over a compact convex set. Let us first introduce a few definitions.
is convex for all λ ∈ R.
Note that any function that is concave is also quasi-concave; however, the converse is not true. For example, f (x 1 , x 2 ) = x 1 x 2 is a quasi-concave function in R 2 + but is not concave [1]. Therefore, quasi-concave functions are a generalization of concave functions.
Concave minimization is NP-hard even for the special case of minimizing a concave quadratic function over a hypercube [8]. In fact, Mittal and Schulz [20] show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. Since the general concave minimization problem is hard, special cases of the problem have been of interest. To discuss the special cases, we first introduce the notion of a rank of a function following [15], and [24]. Definition 1.2 The rank of a function f : R n → R is the smallest integer k such that, for some function g : R k → R and linearly independent vectors a 1 , a 2 , . . . , a k ∈ R n .
Below are some examples to illustrate the notion of the rank of a function.
where a 1 and a 2 are linearly independent, then f has rank two.
where k ≥ 1 and a 1 , . . . , a k are linearly independent, then f has rank k. 3 x where a 1 and a 2 are linearly independent and a T 2 x = 0, then f has rank two.
The problem of minimizing low-rank concave functions has been extensively studied in the literature. While one might think that the special case of constant-rank functions are easy to optimize, Pardalos and Vavasis [23] show that even minimizing a quadratic function of rank two over a polytope is NP-hard. Matsui [19] proves an even stronger version where he shows that minimizing the product of two strictly positive linear functions over a polytope is NP-hard. Therefore, it is natural to study algorithms that compute near-optimal solutions. This has received considerable interest in the literature both from a point of view of designing computationally efficient heuristics as well as designing polynomial time algorithms with provable performance bounds. Konno et al. [14] consider the low-rank concave quadratic minimization problem and propose a tabusearch heuristic to solve the problem. Porembski [24] considers a generalization of the low-rank concave quadratic functions where the objective function is non-linear for a small number of variables and propose a cutting plane solution approach. Several other solution approaches including enumeration (see for instance [2] and [5]) and cutting plane methods [28] have been studied in the literature. We refer the reader to surveys by Horst and Pardalos [10] and Pardalos and Rosen [22] for an extensive discussion. The above solution approaches are efficient heuristics but do not provide any bounds on the running time or performance of the algorithm.
There has been extensive work from the perspective of designing efficient algorithms with provable performance bounds as well. Vavasis [29] gives an approximation scheme for low-rank quadratic optimization problems. However, the notion of approximation algorithm is different from the one we use in this paper. In [29], a solutionx is an -approximate solution for the problem of minimizing a function f if where M = max x f (x). Therefore, any solution is 1-approximation according to this definition. On the other hand, we consider the standard notion of relative approximation ratio where a solutionx is a (1 + )- Kern and Woeginger [12] consider the problem of minimizing a product of two linear functions and give an FPTAS for the problem. Recall that product of two linear functions is a quasi-concave function of rank 2.
Kelner and Nikolova [11] and Mittal and Schulz [20] are two recent papers that are most closely related to our work. Kelner and Nikolova [11] propose an approximation scheme with polynomial smoothed complexity for the low-rank quasi-concave minimization problem if the objective function satisfies a Lipschitz condition with respect to the L 1 -norm with a polynomially bounded coefficient, i.e., where φ(·) is a fixed polynomial and n is the dimension of the decision space. They also show that it is NP-hard to approximate the general quasi-concave minimization problem by a ratio better than Ω(log n) unless P =NP. In our results, we do not require a Lipschitz condition on the objective but we do assume that the objective function satisfies certain scalability properties described later. The two assumptions are similar in spirit but are independent. It is possible that a function satisfies our assumption of scalability but not the Lipschitz condition of [11] and vice-versa. Moreover, our algorithm is a deterministic polynomial time algorithm and is based on solving several linear minimization problems with appropriate objectives that are based on the gradient of the original function. This approach provides geometric insights for the concave and quasi-concave minimization that are of independent interest and can be useful in designing practical algorithms for more general concave minimization problems.
In another recent paper, Mittal and Schulz [20] independently propose an FPTAS for optimizing a class of low-rank functions (not necessarily quasi-concave). Their algorithm is based on considering the projection in the low-rank subspace and solving feasibility problems at the set of appropriately chosen grid points.
The algorithm in [20] works for minimizing more general functions than just quasi-concave; however, it does not give an extreme point solution if the function is quasi-concave. To obtain an extreme point solution for quasi-concave functions, the authors refer to results of Diakonikolas and Yannakakis [6] who construct an approximate Pareto optimal curves efficiently in the context of multi-objective optimization. The problem of constructing an approximate Pareto optimal set for multi-objective optimization is also closely related and has been studied extensively (see Safer and Orlin [27], Papadimitriou and Yannakakis [21]). Moreover, the FPTAS presented in our paper is more efficient for quasi-concave functions under mild assumptions and we compare the running time in Section 2. Also, as mentioned earlier, our algorithm is based on solving a polynomial number of appropriately chosen linear optimization problems over a convex set and this technique could be of independent interest.

Model and Assumptions
We consider the problem of minimizing a quasi-concave function f : R n → R + of rank k over a compact convex set P ⊂ R n . We assume that the convex set P is specified by a separation oracle. Since f has rank k, there exists g : R k → R + and a 1 , . . . , a k ∈ R n such that We assume that the linearly independent vectors a 1 , . . . , a k that define the low-rank representation of f are given. Also, the function g and gradient ∇g are given as a black-box input. These can be readily computed for a large class of quasi-concave functions f . In addition, we assume that g satisfies the following conditions.
(P1) The gradient ∇g ≥ 0 for all y ∈ R k + , i.e. partial derivative for all α ≥ 1 and some constant c.
The second assumption is similar in spirit to the Lipschitz condition assumed in [11]. The above conditions are satisfied by a large class of quasi-concave functions that include linear multiplicative functions [16] and utility functions from microeconomics theory such as Leontief utilities and Cobb-Douglas utilities [7]. We can now state our minimization problem, Π, as follows.
The following result about the minimum of quasi-concave functions is well known (for instance, see Bertsekas et al. [4]). In this paper, we present an FPTAS for the above problem where k is a constant and g satisfies (P1), (P2) and (P3) and relies on the above proposition. The basic idea of the algorithm is as follows. For a polynomial number of values of λ ∈ R, we consider level curves, U λ of g, where For each level curve, we solve a polynomial number of linear optimization problems with appropriate objective functions that depend on the gradient of the function on the level curve. Using the properties of quasi-concavity and the fact that g satisfies (P1), (P2), and (P3), we prove that considering a polynomial number of different values of λ is sufficient to find a near-optimal solution to Π that is also an extreme point of P . Our results can also be extended to the problem of maximizing low-rank quasi-convex functions under similar assumptions.
Outline. In Section 2, we describe our FPTAS for minimizing constant rank quasi-concave functions. We discuss applications of our algorithm to combinatorial problems in Section 3.

(1 + )-Approximation Algorithm
We present a fully polynomial time approximation scheme (FPTAS) for the problem Π (1.2) in this section. Since f has rank k and can be expressed using g : R k → R using an affine transformation through vectors a 1 , . . . , a k , we can reformulate Π as follows. Let and Since we have a separation oracle for P and A is known, we also have a separation oracle for P k . Now, Π can be equivalently expressed as, min g(y) Note that P k ⊂ R k + since P ⊂ R n + and a 1 , . . . , a k ∈ R n + . From any solution, y ∈ P k , we can construct a solution x ∈ P by solving a linear feasibility problem. We first show that g is also a quasi-concave function.
Lemma 2.1 Let the function g be as defined in (1.1). If the function f is quasi-concave, then g is also quasi-concave.
Proof: Consider any λ ∈ R and consider the upper level set, Consider any y 1 , y 2 ∈ U g λ and 0 ≤ α ≤ 1 and letŷ = αy 1 + (1 − α)y 2 . We need to prove thatŷ ∈ P k . There exist x 1 , x 2 ∈ P such that g(y j ) = f (x j ) ≥ λ and y j = A T x j , j = 1, 2. Let Therefore,ŷ ∈ P k which implies that g is quasi-concave.
Let us introduce a few notations. For all i = 1, . . . , k, Note that Assumption (P3) implies that i > 0 for all i = 1, . . . , k. Let and Clearly, P k ⊆ H k . Note that | log i | and | log u i | are both polynomial in the input size for all i = 1, . . . , k. Therefore, we can construct a polynomial size grid to discretize the set H k−1 . In particular, for a given > 0, let Consider the following set Γ ⊂ H k−1 .
Note that for any point (y 1 , . . . , y k−1 ) ∈ H k−1 , there exists (ỹ 1 , . . . ,ỹ k−1 ) ∈ Γ such that y j ≤ (1 + )ỹ j for all j = 1, . . . , k − 1. Therefore, H k−1 can be approximated by the discrete set Γ . Now, we can describe the basic algorithm as follows. Let z * denote the optimal objective value of (2.2). We maintain an upper bound, z U and a lower bound, z L on z * and let In each iteration of the algorithm, we solve a polynomial number of linear optimization problems over P k and either update the upper bound to less than or equal to z or update the lower bound to z/(1 + O( )). In particular, we show that one of the following holds in each iteration of the algorithm.
1. Either the algorithm finds a feasible solution to Π with objective value less than or equal to z, or 2. z * ≥ z 1+O( ) . Therefore, in each iteration we are able to reduce the ratio between the upper and lower bound by a constant factor and the algorithm terminates when z U and z L are close. So the number of iterations in the algorithm is O log z U z L . We give a formal description of the algorithm below. For any λ ∈ R, let To compute T λ for any λ, > 0, we need to solve a single variable equation to compute v k such that g(v 1 , . . . , v k ) = λ given any (v 1 , . . . , v k−1 ) ∈ Γ . We assume that this inverse function is given as a black-box input.
Input: A rank k quasi-concave function f , a compact convex set P ⊆ R n + , and > 0.

Initialize
(b) For each v ∈ Tˆ λ , i. Solve the following linear minimization problem LP(v): and let y * v ∈ P k be an extreme point optimal solution for LP(v).  In the following lemmas, we prove the correctness of the algorithm. We first show that z U and z L are initialized correctly. Next, we show that the algorithm updates z L and z U correctly in each iteration. Finally, we show that in each iteration, the algorithm either updates z U or z L , and, therefore, reduces the ratio of z U and z L by a constant factor.

Lemma 2.2 In
Step 2 of Algorithm A, the upper and lower bounds are initialized correctly, i.e., Proof: Note that y i , y u ∈ P k for all i = 1, . . . , k. Therefore, z * ≤ min{g(y i ), g(y i u )} for all i which implies From (2.3), we know that for any y ∈ P k , y i ≥ i for all i = 1, . . . , k. Also, ∇g ≥ 0 for all y ∈ R k + . Therefore, g( 1 , . . . , k ) ≤ g(y), ∀y ∈ P k , which implies g( 1 , . . . , k ) ≤ z * Next, we show that z U and z L are correctly updated by the algorithm.
Proof: Note that z U is updated in Step 3(b)ii only if we find a feasible solution y ∈ P k such that z U > g(y). Therefore, the algorithm updates z U correctly Lemma 2.4 Algorithm A correctly updates z L in Step 3c.
Proof: In any iteration with parameter λ, where z * is the optimal value of (2.2). Suppose y * ∈ P k is an optimal solution of (2.2). By construction of Let v k be such that g(v 1 , . . . , v k ) = λ and v = (v 1 , . . . , v k ) ∈ Tˆ λ . Recall that for all i = 1, . . . , k, g y i (v) denotes the partial derivative of g with respect to y i at v.
where (2.8) follows from the optimality of z * v for LP(v) and (2.9) follows from the fact the g y Here, the last inequality follows from the fact that g(αy) ≤ α c g(y), ∀α ≥ 1 and some constant c Figure 2 illustrates the updates of lower and upper bounds in each iteration. Now, to complete the proof of the correctness, we show that in each iteration, we either update the upper bound or the lower bound.
Lemma 2.5 At the end of each iteration with λ = (z U + z L )/2 in the algorithm, one of the following holds.
1. Either z U is updated to less than or equal to λ, or, 2. z L is updated to λ · (1 +ˆ ) −c in step (3c).
Proof: If z * v ≥ (∇g(v)) T v for all v ∈ Tˆ λ , then from Lemma 2.4, we know that z L is updated to λ·(1+ˆ ) −c . Suppose this is not the case. Then, We show that we can update the upper bound z U in this case. Consider the upper level set Note that v ∈ U λ . Since g is quasi-concave, U λ is convex. Therefore, is a valid supporting hyperplane for U λ [26]. In fact, it is a tangential hyperplane to the level curve {y ∈ R k Therefore, the upper bound z U is updated to g(y * v ) which is less than λ. The following lemma bounds the running time of the algorithm. Proof: The algorithm does binary search for the objective value between the initial upper and lower bounds (z U and z L ) until z U ≤ z L (1 + ). In each iteration of the algorithm, the ratio of the upper and lower bounds reduces by at least (1 −ˆ /2). If z 0 U (z 0 L ) denote the initial upper (lower) bound on the objective value, then the number of iterations is For each iteration with λ = (z L + z U )/2, we solve O(|Tˆ λ |) linear minimization problems over the convex set P . Let Note that log R is polynomial in the input size of the problem since both | log u i | and | log i | are polynomial in input size for all i = 1, . . . , k. Now,

This shows that Algorithm
convex optimization problems (in fact linear optimization over a convex set) which is polynomial in the input size and 1 for a constant k.
In contrast, the running time of the algorithm in [20] depends on (log R) k . We would like to note that we assume ∇g is given as a black-box input and the single-variable inverse function to compute T λ is also given as a black-box input that are not required inputs for the algorithm in [20]. However, for a large class of functions f , ∇g and the inverse function are readily available.
Algorithm A returns y A ∈ P k that is a (1 + )-approximation for minimizing g over P k for any > 0 where y A is an extreme point of P k . To obtain a solution x ∈ P that is an extreme point near-optimal minimizer of f in P , we can solve the optimization problem over P .
min 0 Thus, we have the following theorem.
Theorem 2.7 There is a fully polynomial time approximation scheme (FPTAS) for minimizing a quasiconcave function f of constant rank k over a convex set if f satisfies (P1), (P2) and (P3). Furthermore, the algorithm returns a solution that is an extreme point of the convex set. The running time is polynomial in the input size and 1/ .

Applications to 0-1 Combinatorial Problems
In this section, we consider the following combinatorial problem.
where S ⊆ {0, 1} n and f : R n + → R + is a quasi-concave function of rank k that satisfies assumptions (P1), (P2), and (P3). We further assume that we know the description of the convex hull of S and we can solve linear optimization problems over conv(S) in polynomial time. Our algorithm A directly applies to obtaining a fully polynomial time approximation scheme for (3.1) since A computes a solution that is an extreme point of the conv(S) which belongs to S. Therefore, our result applies when S is the set of s-t paths, spanning trees, perfect matchings or s-t cuts in an undirected graph. Consider an undirected graph G = (V, E), with costs c 1 : E → R + , and c 2 : E → R + . As an example, we obtain an FPTAS for the following problems.
1. Spanning Trees. Let S ⊂ E be the set of spanning trees of G. Algorithm A can be adapted to obtain an FPTAS for min T ∈S c 1 (T ) · c 2 (T ).

2.
Minimum Product Cut. Let S ⊂ E be the set of s-t cuts in G. Algorithm A can be adapted to obtain an FPTAS for min X∈S c 1 (X) · c 2 (X).
3. Minimum Product Path. Let S ⊂ E be the set of s-t paths in G. Algorithm A can be adapted to obtain an FPTAS for min X∈S c 1 (X) · c 2 (X).
We would like to note that for the minimum product cut and minimum product path problems, we can optimize over the dominant of conv(S) instead of conv(S) since the costs are non-negative. The above problems can be generalized to minimizing a product of a constant number of linear costs instead of just two. This generalizes the result of Kern and Woeginger [12] and Goyal et al. [9] who consider minimizing the product of two linear functions over a polytope and give an FPTAS for the same. The general algorithm in [20] for low-rank minimization is not applicable for the above combinatorial problems as the solution is not necessarily an extreme point.

Conclusions
In this paper, we consider minimizing a class of low-rank quasi-concave functions over a convex set and obtain an FPTAS for the problem. Since f is minimized at an extreme point of the convex set, there exists a linear objective function that is minimized at the same extreme point as f . Our algorithm can be interpreted as an efficient search of such a linear objective function where we search over the gradient of f at a polynomial number of points. This search procedure is very general and it would be interesting to see if the idea can be extended to other non-convex optimization problems. In comparison to [20], it gives better running times (O((log R) k−1 ) versus O((log R) k ) of [20]). Furthermore, it is also applicable to providing integer solutions of good quality to combinatorial problems with linear descriptions of their convex hulls as outlined in Section 3, whereas the general algorithm in [20] do not apply to this setting as their solution is not necessarily an extreme point.