On polyhedral approximations of the second-order cone Aharon Ben-Tal 1 and Arkadi Nemirovski 1 May 30, 1999

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1 On polyhedral approximations of the second-order cone Aharon Ben-Tal 1 and Arkadi Nemirovski 1 May 30, 1999 Abstract We demonstrate that a conic quadratic problem min x e T x Ax b, A l x b l 2 c T l x d l, l = 1,..., m }, y 2 = y T y, (CQP) is polynomially reducible to Linear Programming. We demonstrate this by constructing, for every ɛ (0, 1 2 ], an LP program (explicitly given in terms of ɛ and the data of (CQP)) min e T x P x,u ( ) } x + p 0 u (LP) with the following properties: (i) the number dim x + dim u of variables and the number dim p of constraints in (LP) do not exceed [ ] m O(1) dim x + dim b + dim b l ln 1 ɛ ; (ii) every feasible solution x to (CQP) can be extended to a feasible solution (x, u) to (LP); (iii) if (x, u) is feasible for (LP), then x satisfies the ɛ-relaxed constraints of (CQP), namely, Ax b, A l x b l 2 (1 + ɛ)[c T l x d l ], l = 1,..., m. 1 Introduction The initial motivation for the question posed and resolved in this paper originated from the practical need to solve numerically a conic quadratic problem } min e T x Ax b, A l x b l 2 c T x l x d l, l = 1,..., m, (CQP) y 2 = y T y being the Euclidean norm. There are two major sources of recent interest in these problems: (CQP) is a natural form of several important applied problems, e.g., problems with Coulomb friction in contact mechanics, see [3, 5, 6, 10]; Conic quadratic constraints have very powerful expressive abilities, which allows one to cast a wide variety of nonlinear convex optimization problems in the form of (CQP), see, e.g., [6, 7] and Examples 1 5 below. 1 Faculty of Industrial Engineering and Management, Technion Israel Institute of Technology, Technion City, Haifa 32000, Israel Funded by the Germany-Israeli Foundation for R&D contract No. I /95 and the Israel Ministry of Science grant #

2 As a matter of fact, the case of (CQP) is covered by the existing general theory of polynomial time interior point (IP) methods (e.g., the general theory for well-structured convex programs [7] and the Nesterov-Todd theory of IP algorithms for conic problems on self-scaled cones [8, 9]). These theories yield both algorithms and complexity bounds; in view of these bounds, (CQP) with m constraints and n variables is not more difficult than a linear programming problem of similar sizes (in both cases, the problem can be solved within an accuracy δ in O(1) m ln(o(δ 1 )) steps of a polynomial time IP method, with no more than O(n 2 (m + n)) arithmetic operations per step). Moreover, there exists software implementing the theoretical schemes. However, to the best of our knowledge the IP software for (CQP) available at the moment, although capable of handling problems with tens of thousands of conic quadratic constraints, imposes severe restrictions on the design dimension of the problem (a few thousand variables). In this respect the state of the art in numerical processing of CQP s is incomparable to that in LP, where we can solve routinely problems with even hundreds of thousands of variables and constraints. Given this huge difference, the question arises: can we process CQP s via LP techniques? Mathematically, the question is whether we can approximate a CQP problem by an LP one, without increasing dramatically the sizes of the problem, and this is the question we address in this paper. We start with a precise formulation of the question. Let ɛ > 0, and let L k = (y, t) R k R t y 2 } be the (k + 1)-dimensional Lorentz cone. We define a polyhedral ɛ-approximation of L k as a linear mapping Π k (y, t, u) : R k R R p k R q k such that (i) If (y, t) L k, then there exists u R p k such that Π k (y, t, u) 0; (ii) If (y, t) R k R is such that Π k (y, t, u) 0 for some u, then y 2 (1 + ɛ)t. Geometrically: the system of homogeneous linear constraints Π k (y, t, u) 0 defines a polyhedral cone K in the space of (y, t, u)-variables; we say that Π k ( ) is a polyhedral ɛ-approximation of L k, if the projection of K on the space of (y, t)-variables contains L k and is contained in the (1+ɛ)-extension of L k in the image of L k under linear mapping (x, t) (x, (1+ɛ) 1 t) (note that for ɛ close to 0 the mapping is nearly the identity ). Let k l, l = 1,..., m, be the row sizes of the matrices A l in (CQP). Given polyhedral ɛ- approximations Π k l( ) of the Lorentz cones L k l, l = 1,..., m, we can approximate (CQP) by the linear programming problem ( min e T x Ax b, Π k l A l x b l, c T x,u l } m l x d l, u l) } 0, l = 1,..., m. (LP) From the definition of a polyhedral approximation it follows that (CQP) and (LP) are linked as follows: both problems have the same objectives, and the projection ˆX of the feasible set of (LP) onto the x-space is in between the feasible set of (CQP) and that of its the ɛ-relaxation: } min e T x Ax b, A l x b l 2 (1 + ɛ)[c T x l x d l ], l = 1,..., m. (CQP ɛ ) We see that if ɛ is close to 0, then (LP) is a good approximation of (CQP). 2

3 Note that the sizes of (LP) are larger than those of (CQP); specifically, the design dimension of (LP) is m N = n + [n = dim x], p kl and the total number of linear constraints in (LP) is M = k 0 + m q kl, where k 0 is the row size of A; recall that p k and q k are the dimension of the u-vector and the image dimension of the polyhedral approximation Π k (,, u), respectively. The weakest necessary requirement for the proposed scheme to be computationally meaningful is that the size M + N of (LP) should be polynomial in the size n + m k l of (CQP) and should grow moderately as ɛ l=0 approaches 0. We arrive at the following question: (?) Can we construct a polyhedral ɛ-approximation of L k with the sizes p k + q k growing moderately as k grows and as ɛ approaches 0? The straightforward approach when L k is approximated by a circumscribed polyhedral cone with a sufficiently large number of facets does not work: the required number of facets blows up exponentially as k grows (even with ɛ = 1, a rough lower bound on the number of required facets is expk/8}). Surprisingly, the situation is not as bad as suggested by the latter observation. We prove the following result: Theorem 1.1 For every k and every ɛ (0, 1], L k admits a polyhedral ɛ-approximation with p k + q k O(1)k ln 2 ɛ (1) (from now on, all O(1) s are absolute constants). Theorem 1.1 establishes an interesting (and, to the best of our knowledge, new) geometric fact and as such seems to be important by its own right. As far as the computational consequences are concerned, the situation is as follows. It was already mentioned that theoretically problem (CQP) with n variables and m constraints is not more difficult than a linear programming problem of the same sizes. When replacing (CQP) with its polyhedral approximation, we increase both m and n, thus losing in the complexity. Note, however, that theoretical complexity analysis and computational practice are not the same: performance and reliability of modern commercial LP software are much better than those of CQP-solvers available at the moment, and this (hopefully, temporary) difference can make the use of polyhedral approximation practically meaningful, especially in the case when this approximation does not increase that dramatically the sizes of the problem. The latter happens, e.g., when the row sizes k l of the conic quadratic constraints in (CQP) are small (recall that we need O(k l ln(ɛ 1 )) linear constraints and extra variables to approximate the constraint A l x b l 2 c T l x d l). In this respect it should be mentioned that there are important sources of conic quadratic problems with fairly small k l s, e.g., problems with Coulomb friction (k l 3) [3, 5, 6, 10], and truss topology design problems (k l 2) [1]. We list below several other examples where the polyhedral approximation may be of practical use. 3

4 Example 1: Quadratically constrained convex quadratic program min x x T Q T 0 Q 0 x 2b T 0 x : x T Q T i Q i x 2b T i x c i, i = 1,..., m } can be posed equivalently as the conic quadratic program ( ) Q0 x min t : t + 2bT 0 x + 4 ( ) Qi x, 2bT i x + c } i + 4, i = 1,..., m. x,t t+2b T 0 x 4 4 2b T i x+c i 4 4 If the number of truly quadratic (i.e., with Q i 0) constraints is relatively small, a polyhedral approximation of these constraints does not increase significantly the sizes of the problem. We see that linear programming software can be easily adapted to handle mixed linearly-quadratically constrained convex quadratic problems with few quadratic constraints. The accuracy of the resulting approximation, as well as those of the approximations in Examples 2 4 below, will be discussed in Section 4. Example 2: Geometric means. with n π i 1. The hypograph i=1 Let π i = p i p, p i, p N +, be positive rationals, i = 1,..., n, of the concave function f(x) = H = (x, t) R n+1 x 0, t n i=1 n i=1 x π i i } x π i i : R n + R can be represented via conic quadratic inequalities. Indeed, let us choose the smallest integer k such that 2 k p and consider the following system of constraints in variables y = (x, t, τ, y l j }): y 0 (y1 0,..., y0 ) = (x 2 k 1,..., x } 1,..., x } n,..., x n, τ,..., τ, 1,..., 1), }}}}}} p 1 p n 2 k p n y 0 0, p p i 1 0 yj l y2j 1 l 1 yl 1 2j, j = 1,..., 2 k l, l = 1,..., k 1, 0 τ y1 k 1 y2 k 1, t τ. (2) Note that in fact (2) is a system of conic quadratic inequalities, since ( ) } u, v, w 0 : u vw = u, v, w 0 : u v w 2 2 v + w 2 Moreover, it is immediately seen that the projection of the solution set X of (2) on the x, t-space is exactly the set H. Replacing every conic quadratic constraint in (2) by its tight polyhedral approximation, let us denote by Y the solution set of the resulting system of linear inequalities. This set lives 4 }.

5 in a certain R N which contains, as a subspace, the space of y-variables, and the projection of Y on this space is a tight outer approximation of X. It follows that the projection of Y on the subspace of x, t-variables is a tight approximation of the hypograph H of f. As a result, we get the possibility to process via LP optimization programs with linear objectives and constraints of the form a j x π lj l b [a j 0, π lj 1, π lj are positive rationals]. j l l Example 3: Inverse geometric means. i = 1,..., n. The epigraph Let π i = p i p, p i, p N +, be positive rationals, of the function f(x) = n i=1 E = (x, t) R n+1 x > 0, t choosing the smallest integer k such that 2 k constraints in variables y = (x, t, y l j }): n i=1 x π i i } x π i i can be represented via conic quadratic inequalities. Indeed, p + i p i, consider the following system of y 0 (y1 0,..., y0 ) = (x 2 k 1,..., x } 1,..., x } n,..., x n, t,..., t, 1,..., 1 ), }}}}}} p 1 p n p p i 2 k p i y 0 0, 0 yj l y2j 1 l 1 yl 1 2j, j = 1,.., 2 k l, l = 1,..., k 1, 1 y1 k 1 y2 k 1. (3) As in (2), the system (3) is in fact a system of conic quadratic inequalities; it is immediately seen that the projection of the solution set X of this system onto the space of x, t-variables is exactly the epigraph E of f. As in Example 2, we can use tight polyhedral approximations of the conic quadratic inequalities in (3) to get a tight polyhedral approximation of the epigraph of f. Thus, we get the possibility to process via linear programming optimization programs with linear objectives and constraints of the form a j x π lj l b [a j 0, π lj are positive rationals]. j l Example 4: Geometric programming. To process Geometric Programming problems via LP, it suffices to build a tight polyhedral approximation of the epigraph of the exponential function. Assume that the exponents a T i x of the exponential monomials expat i x} arising in the problem vary in a given segment R a T i x R (R < ). Note that one can enforce this assumption by adding to the original problem the constraints a T i x R, and that that from the computational viewpoint these constraints are indeed a must : e.g., a SUN computer believes that exp750} =, and exp 750} = 0, so that in actual computations one can safely set R = 750. We come to the problem as follows: 5

6 Given a set E R = (x, t) R 2 x R, expx} t}, and an ɛ > 0, find an (1 + ɛ)-polyhedral approximation of the set, i.e., a system P (x, t, u) 0 of affine constraints in variables x, t and additional variables u such that (i) If (x, t) E R, then there exists u such that P (x, t, u) 0; (ii) If (x, t, u) E R, then x R and expx} (1 + ɛ)t. A reasonably good solution to this problem is given by Theorem 1.1. First we observe that E R can be approximated by the following system of quadratic constraints: 1 4 x R, ) u 1, ( 2 q x (1 + 2 q x) 2 + (u 2 1) 2 u [ (1 + 2 q x) 2 u 2 ], u u2 2 u 3, 4u 2 2+i + (u 3+i 1) 2 u 3+i + 1, [ u 2 2+i u 3+i] ( with properly chosen q = O variables u 1 and u 2, reduces to ln R ɛ i = 1,..., q, u 3+q t ). Indeed, the system in question, after eliminating the x R, g q (x) 1 + (2 q x) + (2 q x) (2 q x) (2 q x) 4 24 u 3, u 2 2+i u 3+i, i = 1,..., q, u 3+q t, (4) and the projection of the solution set of the latter system on the plane of (x, t)-variables is the set E q R = (x, t) R2 x R, g (2q ) q (x) t}; which, for large q, is a good approximation of E R due to g (2q ) q (x) (exp2 q x}) (2q) = expx}. Replacing the conic quadratic constraints in the system (4) by their tight polyhedral approximations, given by Theorem 1.1, one obtains a tight polyhedral approximation of E R. Example 5: Robust counterpart of uncertain LP programs with ellipsoidal uncertainty (see [2]). This is a conic quadratic program obtained from an original LP program by replacing every inequality constraint affected by data uncertainty with a conic quadratic constraint. The motivation behind this procedure implies that the stabilizing effect of the procedure remains essentially the same when the conic quadratic constraints are further replaced with their polyhedral ɛ-approximations with a moderate (not necessary close to 0) value of ɛ, 6

7 say, ɛ = 1, thus allowing a moderate growth of the number of constraints, as compared to the original uncertain LP problem. The rest of the paper is organized as follows. The simple construction of a concrete polyhedral approximation leading to Theorem 1.1 is presented in the Section 2. In Section 3 we demonstrate that the upper bound in (1) is, in a sense, the best possible. Section 4 contains some concluding remarks. 2 The construction Let ɛ (0, 1] and a positive integer k be given. We intend to build a polyhedral ɛ-approximation of the Lorentz cone L k. Without loss of generality we may assume that k is an integer power of 2: k = 2 θ, θ N. Tower of variables. constraint The first step of our construction is to represent a conic quadratic y y2 k t (CQI) of dimension k + 1 by a system of conic quadratic constraints of dimension 3 each. Namely, let us call the original y-variables variables of generation 0 and let us split them into pairs (y 1, y 2 ),..., (y k 1, y k ). We associate with each pair its successor an additional variable of generation 1. We split the resulting 2 θ 1 variables of generation 1 into pairs and associate with every pair its successor an additional variable of generation 2, and so on; after θ 1 steps we end up with two variables of the generation θ 1. Finally, the only variable of generation θ is the variable t from (CQI). To introduce convenient notation, let us denote by yi l the i-th variable of generation l, so that y1 0,..., y0 k are our original variables y-variables y 1,..., y k, y1 θ t is the original t-variable, and the parents of yi l are the two variables yl 1 2i 1, yl 1 2i. Note that the total number of all variables in this tower of variables is 2k 1. It is clear that the system of constraints [y l 1 2i 1 ]2 + [y l 1 2i ] 2 y l i, i = 1,..., 2 θ l, l = 1,..., θ, (5) is a representation of (CQI) in the sense that a collection (y1 0 y 1,..., yk 0 y k, y1 θ t) can be extended to a solution of (5) if and only if (y, t) solves (CQI). Moreover, let Π l (x 1, x 2, x 3, u l ), l = 1,..., θ, be polyhedral ɛ l -approximations of the cone L 2 = (x 1, x 2, x 3 ) x x2 2 x 3}. Consider the system of linear constraints in variables y l i, ul i : Π l (y l 1 2i 1, yl 1 2i, y l i, u l i) 0, i = 1,..., 2 θ l, l = 1,..., θ. (6) Writing down this system of linear constraints as Π(y, t, u) 0, where Π is linear in its arguments, y = (y 0 1,..., y0 k ), t = yθ 1, and u is the collection of all ul i, l = 1,..., θ and all yl i, l = 1,..., θ 1, we immediately conclude that Π is a polyhedral ɛ-approximation of L k with ɛ = θ (1 + ɛ l ) 1. (7) 7

8 In view of this observation, we may focus on building polyhedral approximations of the Lorentz cone L 2. A polyhedral approximation of L 2. We are about to show that this is given by the following system of linear inequalities (the positive integer ν is a parameter of the construction): (a) (b) ξ 0 x 1 η 0 x 2 ( ) ξ j = cos π ξ j 1 + sin 2 j+1 ( ) η j sin π ξ j 1 + cos 2 j+1 ( ) π η j 1 2 j+1 ( ) π 2 j+1 η j 1, j = 1,..., ν. (8) (c) ξ ν x 3 ( ) η ν tan π ξ 2 ν ν+1 Note that (8) can be straightforwardly written as a system of linear homogeneous inequalities Π (ν) (x 1, x 2, x 3, u) 0, where u is the collection of the 2(ν + 1) variables ξ j, η j, j = 0,..., ν. Proposition 2.1 Π (ν) is a polyhedral δ(ν)-approximation of L 2 = (x 1, x 2, x 3 ) x x2 2 x 3 } with ( ) 1 1 δ(ν) = ( ) 1 = O cos π 4 ν. (9) 2 ν+1 Proof. We should prove that (i) If (x 1, x 2, x 3 ) L 2, then the triple (x 1, x 2, x 3 ) can be extended to a solution to (8); (ii) If a triple (x 1, x 2, x 3 ) can be extended to a solution to (8), then (x 1, x 2 ) 2 (1+δ(ν))x 3. Proof of (i): Given (x 1, x 2, x 3 ) L 2, let us set ξ 0 = x 1, η 0 = x 2, thus ensuring (8.a). Note that (ξ 0, η 0 ) 2 = (x 1, x 2 ) 2 and that the point P 0 = (ξ 0, η 0 ) belongs to the first quadrant. Now, for j = 1,..., ν let us set ( ) ( ) ξ j = cos π ξ j 1 + sin π η j 1 2 j+1 ( ) 2 j+1 ( ) η j = sin π ξ j 1 + cos π η j 1, 2 j+1 2 j+1 thus ensuring (8.b), and let P j = (ξ j, η j ). The point P j is obtained from P j 1 by the following construction: we rotate clockwise P j 1 by the angle φ j = π, thus getting a point Q j 1 ; if 2 j+1 this point is in the upper half-plane, we set P j = Q j 1, otherwise P j is the reflection of Q j 1 with respect to the x-axis. From this description it is clear that (I) P j 2 = P j 1 2, so that all vectors P j are of the same Euclidean norm as P 0, i.e., of the norm (x 1, x 2 ) 2 ; (II) Since the point P 0 is in the first quadrant, the point Q 0 is in the angle π 4 arg(p ) π 4, so that P 1 is in the angle 0 arg(p ) π 4. The latter relation, in turn, implies that Q1 is in the angle π 8 arg(p ) π 8, whence P 2 is in the angle 0 arg(p ) π 8. Similarly, P 3 is in the angle 0 arg(p ) π 16, and so on: P j is in the angle 0 arg(p ) π. 2 j+1 By (I), ξ ν P ν 2 = (x 1, x 2 ) 2 x 3, so that the first inequality in (8.c) is satisfied. By (II), P ν is in the angle 0 arg(p ) π, so that the second inequality in (8.c) is also satisfied. 2 ν+1 We have extended a point from L 2 to a solution to (8). 8

9 Proof of (ii): Assume that (x 1, x 2, x 3 ) can be extended to a solution (x 1, x 2, x 3, ξ j, η j } ν j=0 ) to (8). Let us set P j = (ξ j, η j ). From (8.a, b) it follows that all vectors P j are nonnegative. We have P 0 2 (x 1, x 2 ) 2 by (8.a). Now, (8.b) implies P j 2 P j 1 2. Thus, P ν 2 (x 1, x 2 ) 2. On the other hand, by (8.c) one has P ν 1 2 ( π )ξ ν ( 1 cos 2 ν+1 π )x 3, so that cos 2 (x ν+1 1, x 2 ) 2 (1 + δ(ν))x 3, as claimed. Proof of Theorem 1.1: Specifying in (6) the mappings Π l ( ) as Π (νl) ( ), we conclude that for every collection of positive integers ν 1,..., ν θ the following system of linear inequalities describes a polyhedral approximation Π ν1,...,ν θ (y, t, u) of L k, k = 2 θ : (a l,i ) (b l,i ) (c l,i ) ξ 0 l,i y2i 1 l 1 ηl,i 0 y l 1 2i ξ j ( ) l,i = cos π ξ j 1 ( ) 2 j+1 l,i + sin π η j 1 ( ) 2 j+1 ( l,i ) η j l,i sin π ξ j 1 2 j+1 l,i + cos π η j 1, 2 j+1 l,i ξ ν l l,i yi l ( ) η ν l l,i tan π ξ ν 2 ν l +1 l l,i i = 1,..., 2 θ l, l = 1,..., θ. The approximation possesses the following properties: j = 1,..., ν l. (10) 1. The dimension of the u-vector (comprised of all variables in (10) except y i = yi 0}k i=1 and t = y1 θ) is θ p(k, ν 1,..., ν θ ) k + O(1) 2 θ l ν l ; 2. The image dimension of Π ν1,...,ν θ ( ) (i.e., the # of linear inequalities plus twice the # of linear equations in (10)) is 3. The quality β of the approximation is Given ɛ (0, 1] and setting q(k, ν 1,..., ν θ ) O(1) β = β(ν 1,..., ν θ ) = θ 2 θ l ν l ; θ 1 ( ) 1. cos π 2 ν l +1 ν l = O(1)l ln 2, l = 1,..., θ, ɛ with properly chosen absolute constant O(1), we ensure that as claimed in Theorem 1.1. β(ν 1,..., ν θ ) ɛ, p(k, ν 1,..., ν θ ) O(1)k ln 2 ɛ, q(k, ν 1,..., ν θ ) O(1)k ln 2 ɛ, 9

10 Simplifying the polyhedral approximation. It is easily seen that the approximation (10) can be economized by reducing the image dimension q k = q(k, ν 1,..., ν θ ) and the number p k = p(k, ν 1,..., ν θ ) of extra variables without influencing the quality of the approximation. Specifically, (A) The variables ξl,i 1,..., ξν l l,i in (10) are given by linear equations; one can use these equations to represent ξ j l,i, j 1, as linear combinations of ξ0 l,i and the η l,i-variables, thus eliminating the variables ξl,i 1,..., ξν l l,i along with the corresponding linear equations; (B) For l = 2,..., θ, one can replace inequalities (10.a l,i ) with ξ 0 l,i y l 1 2i 1, η0 l,i y l 1 2i, (11) since the system we end up with after this modification ensures the nonnegativity of yi l 1, l = 2,..., θ; (C) In the system of linear inequalities obtained from (10) after modifications (A) and (B), every variable yi l, l = 1,..., θ 1, enters the inequalities (c l,i ) : ξ ν l l,i yl i, (11) : y l i ξ 0 l+1,(i+1)/2, i odd η 0 l+1,i/2, i even ; (12) it is immediately seen that we preserve the approximation property and the quality of approximation by converting these inequalities into equations. After this modification, we can eliminate from the system of constraints the inequalities (11), and eliminate the variables ξ 0 l,i and η0 l,i, l = 2,..., θ (these variables become linear forms of the remaining ones). To get an impression of the complexity of our polyhedral approximation scheme, we present a table showing the numbers of extra variables p k and linear constraints q k in a polyhedral ɛ-approximation of a conic quadratic constraint y 2 t with dim y = k: ɛ = 10 1 ɛ = 10 3 ɛ = 10 6 ɛ = k p k q k p k q k p k q k p k q k 5.2k ln k ln 1 4.7k ln k ln 1 4.1k ln 1 9.0k ln 1 4.0k ln 1 8.3k ln 1 ɛ ɛ ɛ ɛ ɛ ɛ ɛ ɛ A lower bound We have demonstrated that for every ɛ (0, 1], the Lorentz cone L k for every ɛ (0, 1] admits a polyhedral ɛ-approximation of the size p k + q k O(1)k ln 2 ɛ. We are about to prove that the order of this upper bound in fact cannot be improved: Proposition 3.1 Let k be a positive integer, ɛ (0, 0.5], and let Π(y, t, u) : R k R R p R q 10

11 be a polyhedral ɛ-approximation of L k. Then q O(1)k ln 1 ɛ (13) with positive absolute constant O(1). Proof. The projection L k of the polyhedral cone K = (y, t, u) Π(y, t, u) 0} on the plane of (y, t)-variables does not contain lines; replacing, if necessary, u with its projection on a properly chosen subspace in R p, we may assume that the cone K itself does not contain lines, so that K is a conic hull of its extreme rays. Let N be the number of extreme rays of K; one clearly has N 2 q. Now, since K is the conic hull of N rays, L k is the conic hull of N rays R 1,..., R N, and every one of these rays intersects the hyperplane H = t = 1} in the (y, t)-space. Now let us look at the set G = y (y, 1) L k }. Since Π( ) is a polyhedral ɛ-approximation of L k, the set G contains the unit k-dimensional Euclidean ball B and is contained in the ball (1 + ɛ)b. On the other hand, G is the convex hull of N points y 1,..., y N the y-components of the intersections R i H. It is well-known (see, e.g., [4]) that in the situation in question one has where O(1) is a positive absolute constant. N expo(1)k ln 1 } ɛ 0.5, (14) ɛ To make the presentation self-contained, here is a derivation of (14). We first observe that the N Euclidean balls B i centered at y i of the radius ρ = 2ɛ(1 + ɛ) cover the boundary of (1 + ɛ)b. Indeed, if for some ρ 0 there exists a point y S ((1 + ɛ)b) which is at the distance ρ from every y i, i = 1,..., N, then the convex hull of y 1,..., y N is contained in the convex hull of the set z z ɛ, z y 2 ρ}, so that the latter convex hull contains B; this observation via elementary geometry implies that ρ ρ. Comparing the (k 1)-dimensional areas of the sphere S and the spherical hats B i S, we conclude that the inclusion S B i implies (14). i Combining (14) with the inequality N 2 q, we deduce that q O(1)k ln 1 ɛ, as claimed. 4 Concluding remarks With our approach, we are able to associate with a given conic quadratic ( problem ) (CQP) and a given ɛ (0, 1] an LP program (LP ɛ ) of the same (up to a factor O ln 1 ɛ ) size such that the projection of the feasible set of the latter problem on the x-space is in between the feasible set of (CQP) and the feasible set of the ɛ-relaxation (CQP ɛ ) of (CQP). Thus, (LP ɛ ) is a good approximation of (CQP), provided that the feasible sets of (CQP) and (CQP ɛ ) are close to each 11

12 other. Note that the latter is not necessarily the case; one can easily find an example where (CQP) is infeasible, while all problems (CQP ɛ ), ɛ > 0, are feasible. There is, however, a simple sufficient condition ensuring that the feasible sets of (CQP) and (CQP ɛ ) are O(ɛ)-close to each other: Proposition 4.1 Assume that (CQP) is (i) strictly feasible: there exist x and r > 0 such that A x b, A l x b l 2 [c T l x d l ] r, l = 1,..., m; (ii) semi-bounded : there exists R such that Ax b, A l x b l 2 c T l x d l, l = 1,..., m c T l x d l R, l = 1,..., m. Then for every ɛ > 0 such that γ(ɛ) Rɛ r < 1 one has γ(ɛ) x + (1 γ(ɛ))feas((cqp ɛ )) Feas(CQP) Feas((CQP ɛ )), (15) where Feas((P)) denotes the feasible set of a problem (P). Proof. We already know that the right inclusion in (15) holds true. In order to prove the left inclusion, let y be a feasible solution to (CQP ɛ ); we should prove that the point x = (1 γ)y+γ x is feasible for (CQP). Since both x and y satisfy the linear constraints Ax b, all we should prove is that A l x b l 2 c T l x d l, l = 1,..., m. (16) For δ [0, 1], let x δ = (1 δ)y + δ x. By (i) and since y is feasible for (CQP ɛ ), we have Setting δ = max l A l x b l 2 c T l x d l r, A l y b l 2 (1 + ɛ)[c T l y d l }} t l ], l = 1,..., m. ɛt l r+ɛt l, we conclude that A l x δ b l 2 [c T l x δ d l ] + [(1 δ)ɛt l δr] [c T l x δ d l ], l = 1,..., m, and clearly Ax δ b. Thus, x δ is feasible for (CQP). It follows from (ii) that [c T l x δ d l ] = δ[c T l x d l ] + (1 δ)t l R, l = 1,..., m; since x is feasible for (CQP), we have [c T l x d l] 0, whence (1 δ)t l R for all l. Recalling the origin of δ, we conclude that δ = ɛt r+ɛt for some t satisfying (1 δ)t R. From these relations and in view of γ(ɛ) Rɛ R r < 1 we conclude that t 1 γ(ɛ), whence δ γ(ɛ). Since δ γ(ɛ) 1 and both x δ and x 1 = x are feasible for (CQP), so is x = x γ (ɛ). Proposition 4.1 allows us to quantify the actual quality of polyhedral approximations of conic quadratic problems, in particular, those mentioned in Examples 1 4 of the introduction. In these examples, we are interested in fact in building a polyhedral approximation of the epigraph Epi(f) = (x, t) x Dom f, t f(y)} 12

13 of a particular convex function f(x), i.e., in building a system of linear inequalities P x + tp + Qu + q 0 (S) in such a way that the projection of the solution set of the system onto the (y, t)-space is the epigraph of a certain function f close to f. Specifically, Examples 1 4 deal with the following functions f: Example 1: f(x) = x T Q T Qx : R n R; Example 2: f(x) = n x π i i : R n + R, with π i = p i p, p i, p N +, π i 1; i=1 i Example 3: f(x) = n i=1 x π i i : int R n + R, with π i = p i p, p i, p N + ; Example 4: f(x) = expx}. In all these examples, the closeness between f and f we can efficiently ensure is the uniform closeness on compact subsets of the domain of the target function f. Combining straightforwardly Theorem 1.1, Proposition 4.1 and the approximation schemes mentioned in Introduction in connection with Examples 1 4, we come to the following results: Example 1: for every ɛ (0, 1], R > 0 one can explicitly construct a system (S) of the size Size(S) dim x + dim u + dim q O(n) ln in such a way that the associated function f satisfies (1 + R n)(1 + n max Q ij ) i,j ɛ [n = dim x] f(x) f(x) ɛ (x : x R); Example 2: for every ɛ (0, 1], R > 0) one can explicitly construct a system (S) of the size Size(S) O(p) ln in such a way that the associated function f satisfies p(1 + R) ɛ f(x) f(x) ɛ (x 0 : x R); Example 3: for every ɛ (0, 1], R > 0 one can explicitly construct a system (S) of the size Size(S) O(p + (p + p i )(1 + R) i p i ) ln ɛ i in such a way that the associated function f satisfies f(x) f(x) ɛ (x : R 1 x i R, i = 1,..., n); Example 4: for every ɛ (0, 1], R > 0 one can explicitly construct a system (S) of the size [ Size(S) O(1) R + ln 2 ] (1 + R) ln ɛ ɛ in such a way that the associated function f satisfies f(x) expx} (1 + ɛ) f(x) (x : x R). 13

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