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                                                                    EXHIBIT 4.21


                                 PROMISSORY NOTE


$1,470,320                                                       August 20, 1997


      FOR VALUE RECEIVED, ILOG S.A., a French societe anonyme ("COMPANY"),
promises to pay to CPLEX Optimization, Inc. ("PAYEE") the principal sum of one
million four hundred seventy thousand three hundred twenty dollars, or such
lesser amount as shall then equal the outstanding principal amount hereof,
together with interest from the date of this Promissory Note (the "NOTE") on the
unpaid principal balance at a rate per annum equal to 6.39% (the AFR Rate)
computed on the basis of the actual number of days elapsed and a year of 365
days, with such unpaid interest compounded monthly. Subject to the provisions of
Sections 2(c) and 6 below, all principal not previously paid pursuant to Section
2 hereof, together with any then unpaid and accrued interest, shall be due and
payable at the earlier of (i) the fourth anniversary of the date of this Note or
(ii) when such amounts are declared due and payable by the Payee, or made
automatically due and payable, upon or after the occurrence of an Event of
Default (as defined below).

      This Note is one of three Promissory Notes of the Company (collectively
the "NOTES") of like tenor aggregating U.S. $5,000,000 in principal amount
issued by the Company under an Asset Purchase Agreement dated as of August 4,
1997 (the "ASSET PURCHASE AGREEMENT"), among the Company, ILOG, Inc., the Payee
and the shareholders of the Payee. No Note shall be preferred in any respect
over any other of the Notes because of its sale or transfer by its original or
any subsequent holder.

      The following is a statement of the rights of the holder of this Note and
the conditions to which this Note is subject, and to which the holder hereof, by
the acceptance of this Note, agrees:

            1.    DEFINITIONS. Capitalized terms used without definition herein
which are defined in the Asset Purchase Agreement, dated as of August 4, 1997,
between Company, Todd Lowe, Janet Lowe, Robert Bixby and Payee (the "ASSET
PURCHASE AGREEMENT"), shall have the respective meanings given in the Asset
Purchase Agreement.

            2.    PAYMENTS; PREPAYMENT; RIGHT OF OFFSET.

                  (a)   Scheduled Principal and Interest Payments. On each of
the second and third anniversaries of the date of this Note, Company shall make
a principal payment to Payee in the amount of four hundred ninety thousand one
hundred six dollars and sixty seven cents ($490,106.67). On the fourth
anniversary of the date of this Note, Company shall pay the entire remaining
outstanding principal amount of this Note together with all accrued but unpaid
interest. On the first day of each fiscal quarter of the Company that begins
after the date hereof, the Company shall make a payment to the holder hereof of
all then-accrued and unpaid interest. Each date upon which principal, interest
and/or other amounts are due under any Note is hereinafter defined as a "Due
Date."


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                  (b)   Prepayment. This Note may be prepaid in whole or in part
by the Company at any time, or from time to time, without the further consent of
its holder; provided that all prepayments made by Company on this Note and the
other Notes shall be made on a pro rata basis on all of the Notes in proportion
to the amount outstanding under each such Note. Prepayments shall be first
applied to accrued and unpaid interest, and then to principal. If this Note is
prepaid in part, the principal amount so prepaid shall be applied against the
next scheduled principal payment or payments under Section 2(a).

                  (c)   Company's Right of Offset. As more fully provided in and
subject to the terms of Section 8.5 of the Asset Purchase Agreement, Company
shall under certain circumstances have the right to offset any Damages in
respect of which the Payee or the holder hereof may become obliged to indemnify
the Company or ILOG, Inc. up to a maximum amount equal to eight hundred eighty
two thousand one hundred ninety two dollars ($882,192). Company shall give
notice of any such offset to Payee in the manner set forth in Section 8.4 of the
Asset Purchase Agreement, and interest shall cease to accrue on offset principal
amounts on the date of such notice. The amount offset shall first be applied to
accrued and unpaid interest and then to the principal amount of the next
scheduled principal payment or payments under Section 2(a).

            3.    EVENTS OF DEFAULT. The occurrence of any of the following
shall constitute an "Event of Default" under this Note:

                  (a)   Failure to Pay. Company shall fail to pay any principal,
interest or other payment due under this Note or any other of the Notes on a Due
Date unless such payment is made within five (5) French banking days of
Company's receipt of the holder's written notice to Company of such failure to
pay; or

                  (b)   Voluntary Bankruptcy or Insolvency Proceedings. Company
shall (i) apply for or consent to the appointment of a receiver, trustee,
liquidator or custodian of itself or of all or a substantial part of its
property, (ii) be unable, or admit in writing its inability, to pay its debts
generally as they mature, (iii) make a general assignment for the benefit of its
or any of its creditors, (iv) be dissolved or liquidated in full or in part, or
(v) commence a voluntary case or other proceeding seeking liquidation,
reorganization or other relief with respect to itself or its debts under any
bankruptcy, insolvency or other similar law now or hereafter in effect or
consent to any such relief or to the appointment of or taking possession of its
property by any official in an involuntary case or other proceeding commenced
against it;

                  (c)   Involuntary Bankruptcy or Insolvency Proceedings.
Proceedings for the appointment of a receiver, trustee, liquidator or custodian
of Company or of all or a substantial part of the property thereof, or an
involuntary case or other proceedings seeking liquidation, reorganization or
other relief with respect to Company or the debts thereof under any bankruptcy,
insolvency or other similar law now or hereafter in effect shall be commenced
and an order for relief entered or such proceeding shall not be dismissed or
discharged within sixty (60) days of commencement; or


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                  (d)   Cross Acceleration. Company shall default under any
contract or instrument evidencing indebtedness having a principal amount in
excess of $1,000,000 and such default shall result in such indebtedness becoming
due and owing prior to its stated maturity.

            4.    RIGHTS OF HOLDER UPON DEFAULT. Upon the occurrence or
existence of any Event of Default under Section 3(a) or 3(d) and at any time
thereafter during the continuance of such Event of Default, the holder of this
Note may with the consent of the Majority in Interest, by written notice to
Company, declare all unpaid principal and interest hereunder to be immediately
due and payable without presentment, demand, protest or any other notice of any
kind, all of which are hereby expressly waived. Upon the occurrence or existence
of any Event of Default described in Paragraphs 3(b) or 3(c), immediately and
without notice, all outstanding amounts payable by Company hereunder shall
automatically become immediately due and payable. In addition to the foregoing
remedies, upon the occurrence or existence of any Event of Default, Payee may
exercise any other right, power or remedy permitted to it by law, either by suit
in equity or by action at law, or both. Upon the occurrence and during the
continuance of an Event of Default under Section 3(a), all amounts outstanding
that are past due under this Note shall accrue interest at the rate of 15%. For
purposes of this Section 4, a "MAJORITY IN INTEREST" shall mean holders holding
a majority of the outstanding principal under all of the Notes.

            5.    ASSIGNMENT; SUCCESSORS AND ASSIGNS. This Note may not be
assigned by the Payee without the written consent of the Company and the Company
hereby consents to the assignment of the Note to Janet Lowe. Notwithstanding the
foregoing, the rights and obligations of Company and the Payee of this Note
shall be binding upon and benefit the successors, assigns, heirs, administrators
and transferees of the parties.

            6.    WAIVER AND AMENDMENT. Any provision of the Notes may be
amended, waived or modified only upon the written consent of Company and the
Majority in Interest.

            7.    NOTICES. Any notice, request or other communication required
or permitted hereunder shall be in writing and shall be deemed to have been duly
given if personally delivered or mailed by registered or certified mail, postage
prepaid, or by recognized overnight courier or personal delivery at the
respective addresses of the parties as set forth in the Asset Purchase Agreement
or on the register maintained by Company. Any party hereto may by notice so
given change its address for future notice hereunder. Notice shall conclusively
be deemed to have been given when received.

            8.    PAYMENT. Payment shall be made in lawful tender of the United
States.

            9.    USURY. In the event any interest is paid on this Note which is
deemed to be in excess of the then legal maximum rate, then that portion of the
interest payment representing an amount in excess of the then legal maximum rate
shall be deemed a payment of principal and applied against the principal of this
Note.

            10.   EXPENSES. If action is instituted to collect this Note, the
prevailing party shall be entitled to reimbursement from the other party for all
costs and expenses, including, without limitation, reasonable attorneys' fees
and costs, incurred by the prevailing party in connection with such action.


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            11.   GOVERNING LAW. This Note and all actions arising out of or in
connection with this Note shall be governed by and construed in accordance with
the laws of the State of Delaware, without regard to the conflicts of law
provisions of the State of Delaware or of any other jurisdiction.


            IN WITNESS WHEREOF, Company has caused this Note to be issued as of
the date first written above.

                                       ILOG S.A.


                                       By: /s/ Roger Friedberger
                                           -------------------------------------

                                       Name: Roger Friedberger
                                             -----------------------------------

                                       Title: Chief Financial Officer
                                              ----------------------------------


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